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J. Ambjorn

Publications and source records attributed to J. Ambjorn.

At least 109 records · Page 6Linked to original sources
arXiv2001

Non-perturbative 3d Lorentzian Quantum Gravity

We have recently introduced a discrete model of Lorentzian quantum gravity, given as a regularized non-perturbative state sum over simplicial Lorentzian space-times, each possessing a unique Wick rotation to Euclidean signature. We investigate here the phase structure of the Wick-rotated path integral in three dimensions with the aid of computer simulations. After fine-tuning the cosmological constant to its critical value, we find a whole range of the gravitational coupling constant $k_0$ for which the functional integral is dominated by non-degenerate three-dimensional space-times. We therefore have a situation in which a well-defined ground state of extended geometry is generated dynamically from a non-perturbative state sum of fluctuating geometries. Remarkably, its macroscopic scaling properties resemble those of a semi-classical spherical universe. Measurements so far indicate that $k_0$ defines an overall scale in this extended phase, without affecting the physics of the continuum limit. These findings provide further evidence that discrete {\it Lorentzian} gravity is a promising candidate for a non-trivial theory of quantum gravity.

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arXiv2001

A non-perturbative Lorentzian path integral for gravity

A well-defined regularized path integral for Lorentzian quantum gravity in three and four dimensions is constructed, given in terms of a sum over dynamically triangulated causal space-times. Each Lorentzian geometry and its associated action have a unique Wick rotation to the Euclidean sector. All space-time histories possess a distinguished notion of a discrete proper time. For finite lattice volume, the associated transfer matrix is self-adjoint and bounded. The reflection positivity of the model ensures the existence of a well-defined Hamiltonian. The degenerate geometric phases found previously in dynamically triangulated Euclidean gravity are not present. The phase structure of the new Lorentzian quantum gravity model can be readily investigated by both analytic and numerical methods.

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arXiv2001

On the Spontaneous Breakdown of Lorentz Symmetry in Matrix Models of Superstrings

In string or M theories, the spontaneous breaking of 10D or 11D Lorentz symmetry is required to describe our space-time. A direct approach to this issue is provided by the IIB matrix model. We study its 4D version, which corresponds to the zero volume limit of 4D super SU(N) Yang-Mills theory. Based on the moment of inertia as a criterion, spontaneous symmetry breaking (SSB) seems to occur, so that only one extended direction remains, as first observed by Bialas, Burda et al. However, using Wilson loops as probes of space-time we do not observe any sign of SSB in Monte Carlo simulations where N is as large as 48. This agrees with an earlier observation that the phase of the fermionic integral, which is absent in the 4D model, should play a crucial role if SSB of Lorentz symmetry really occurs in the 10D IIB matrix model.

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arXiv2001

Monte Carlo Studies of the Dimensionally Reduced 4d SU(N) Super Yang-Mills Theory

We simulate a supersymmetric matrix model obtained from dimensional reduction of 4d SU(N) super Yang-Mills theory. The model is well defined for finite N and it is found that the large N limit obtained by keeping g^2 N fixed gives rise to well defined operators which represent string amplitudes. The space-time structure which arises dynamically from the eigenvalues of the bosonic matrices is discussed, as well as the effect of supersymmetry on the dynamical properties of the model. Eguchi-Kawai equivalence of this model to ordinary gauge theory does hold within a finite range of scale. We report on new simulations of the bosonic model for N up to 768 that confirm this property, which comes as a surprise since no quenching or twist is introduced.

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arXiv2000

Computer Simulations of 3d Lorentzian Quantum Gravity

We investigate the phase diagram of non-perturbative three-dimensional Lorentzian quantum gravity with the help of Monte Carlo simulations. The system has a first-order phase transition at a critical value $k_0^c$ of the bare inverse gravitational coupling constant $k_0$. For $k_0 > k_0^c$ the system reduces to a product of uncorrelated Euclidean 2d gravity models and has no intrinsic interest as a model of 3d gravity. For $k_0 < k_0^c$, extended three-dimensional geometries dominate the functional integral despite the fact that we perform a sum over geometries and no particular background is distinguished at the outset. Furthermore, all systems with $k_0 < k_0^c$ have the same continuum limit. A different choice of $k_0$ corresponds merely to a redefinition of the overall length scale.

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arXiv2000

Simulating Simplified Versions of the IKKT Matrix Model

We simulate a supersymmetric matrix model obtained from dimensional reduction of 4d SU(N) super Yang-Mills theory (a 4d counter part of the IKKT model or IIB matrix model). The eigenvalue distribution determines the space structure. The measurement of Wilson loop correlators reveals a universal large N scaling. Eguchi-Kawai equivalence may hold in a finite range of scale, which is also true for the bosonic case. We finally report on simulations of a low energy approximation of the 10d IKKT model, where we omit the phase of the Pfaffian and look for evidence for a spontaneous Lorentz symmetry breaking.

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arXiv2000

Integrable Ladder t-J Model with Staggered Shift of the Spectral Parameter

The generalization of the Yang-Baxter equations (YBE) in the presence of Z_2 grading along both chain and time directions is presented and an integrable model of t-J type with staggered disposition along a chain of shifts of the spectral parameter is constructed. The Hamiltonian of the model is computed in fermionic formulation. It involves three neighbour site interactions and therefore can be considered as a zig-zag ladder model. The Algebraic Bethe Ansatz technique is applied and the eigenstates, along with eigenvalues of the transfer matrix of the model are found. In the thermodynamic limit, the lowest energy of the model is formed by the quarter filling of the states by fermions instead of usual half filling.

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arXiv2000

Monte Carlo Studies of the IIB Matrix Model at Large N

The low-energy effective theory of the IIB matrix model developed by H. Aoki et al. is written down explicitly in terms of bosonic variables only. The effective theory is then studied by Monte Carlo simulations in order to investigate the possibility of a spontaneous breakdown of Lorentz invariance. The imaginary part of the effective action, which causes the so-called sign problem in the simulation, is dropped by hand. The extent of the eigenvalue distribution of the bosonic matrices shows a power-law large N behavior, consistent with a simple branched-polymer prediction. We observe, however, that the eigenvalue distribution becomes more and more isotropic in the ten-dimensional space-time as we increase N. This suggests that if the spontaneous breakdown of Lorentz invariance really occurs in the IIB matrix model, a crucial role must be played by the imaginary part of the effective action.

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arXiv2000

Lattice Gauge Fields and Discrete Noncommutative Yang-Mills Theory

We present a lattice formulation of noncommutative Yang-Mills theory in arbitrary even dimensionality. The UV/IR mixing characteristic of noncommutative field theories is demonstrated at a completely nonperturbative level. We prove a discrete Morita equivalence between ordinary Yang-Mills theory with multi-valued gauge fields and noncommutative Yang-Mills theory with periodic gauge fields. Using this equivalence, we show that generic noncommutative gauge theories in the continuum can be regularized nonperturbatively by means of {\it ordinary} lattice gauge theory with 't~Hooft flux. In the case of irrational noncommutativity parameters, the rank of the gauge group of the commutative lattice theory must be sent to infinity in the continuum limit. As a special case, the construction includes the recent description of noncommutative Yang-Mills theories using twisted large $N$ reduced models. We study the coupling of noncommutative gauge fields to matter fields in the fundamental representation of the gauge group using the lattice formalism. The large mass expansion is used to describe the physical meaning of Wilson loops in noncommutative gauge theories. We also demonstrate Morita equivalence in the presence of fundamental matter fields and use this property to comment on the calculation of the beta-function in noncommutative quantum electrodynamics.

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arXiv2000

Large N Dynamics of Dimensionally Reduced 4D SU(N) Super Yang-Mills Theory

We perform Monte Carlo simulations of a supersymmetric matrix model, which is obtained by dimensional reduction of 4D SU(N) super Yang-Mills theory. The model can be considered as a four-dimensional counterpart of the IIB matrix model. We extract the space-time structure represented by the eigenvalues of bosonic matrices. In particular we compare the large N behavior of the space-time extent with the result obtained from a low energy effective theory. We measure various Wilson loop correlators which represent string amplitudes and we observe a nontrivial universal scaling in N. We also observe that the Eguchi-Kawai equivalence to ordinary gauge theory does hold at least within a finite range of scale. Comparison with the results for the bosonic case clarifies the role of supersymmetry in the large N dynamics. It does affect the multi-point correlators qualitatively, but the Eguchi-Kawai equivalence is observed even in the bosonic case.

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arXiv2000

Nonperturbative Dynamics of Noncommutative Gauge Theory

We present a nonperturbative lattice formulation of noncommutative Yang-Mills theories in arbitrary even dimension. We show that lattice regularization of a noncommutative field theory requires finite lattice volume which automatically provides both an ultraviolet and an infrared cutoff. We demonstrate explicitly Morita equivalence of commutative U(p) gauge theory with (p*n_f) flavours of fundamental matter fields on a lattice of size L with twisted boundary conditions and noncommutative U(1) gauge theory with n_f species of matter on a lattice of size (p*L) with single-valued fields. We discuss the relation with twisted large N reduced models and construct observables in noncommutative gauge theory with matter.

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arXiv2000

Recursive sampling simulations of 3D gravity coupled to scalar fermions

We study numerically the phase structure of a model of 3D gravity interacting with scalar fermions. We measure the 3D counterpart of the "string" susceptibility exponent as a function of the inverse Newton coupling $α$. We show that there are two phases separated by a critical point around $α_c \simeq 2$. The numerical results support the hypothesis that the phase structures of 3D and 2D simplicial gravity are qualitatively similar, the inverse Newton coupling in 3D playing the role of the central charge of matter in 2D.

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arXiv2000

Lorentzian and Euclidean Quantum Gravity - Analytical and Numerical Results

We review some recent attempts to extract information about the nature of quantum gravity, with and without matter, by quantum field theoretical methods. More specifically, we work within a covariant lattice approach where the individual space-time geometries are constructed from fundamental simplicial building blocks, and the path integral over geometries is approximated by summing over a class of piece-wise linear geometries. This method of ``dynamical triangulations'' is very powerful in 2d, where the regularized theory can be solved explicitly, and gives us more insights into the quantum nature of 2d space-time than continuum methods are presently able to provide. It also allows us to establish an explicit relation between the Lorentzian- and Euclidean-signature quantum theories. Analogous regularized gravitational models can be set up in higher dimensions. Some analytic tools exist to study their state sums, but, unlike in 2d, no complete analytic solutions have yet been constructed. However, a great advantage of our approach is the fact that it is well-suited for numerical simulations. In the second part of this review we describe the relevant Monte Carlo techniques, as well as some of the physical results that have been obtained from the simulations of Euclidean gravity. We also explain why the Lorentzian version of dynamical triangulations is a promising candidate for a non-perturbative theory of quantum gravity.

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arXiv1999

On the relation between Euclidean and Lorentzian 2D quantum gravity

Starting from 2D Euclidean quantum gravity, we show that one recovers 2D Lorentzian quantum gravity by removing all baby universes. Using a peeling procedure to decompose the discrete, triangulated geometries along a one-dimensional path, we explicitly associate with each Euclidean space-time a (generalized) Lorentzian space-time. This motivates a map between the parameter spaces of the two theories, under which their propagators get identified. In two dimensions, Lorentzian quantum gravity can therefore be viewed as a ``renormalized'' version of Euclidean quantum gravity.

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arXiv1999

Finite N Matrix Models of Noncommutative Gauge Theory

We describe a unitary matrix model which is constructed from discrete analogs of the usual projective modules over the noncommutative torus and use it to construct a lattice version of noncommutative gauge theory. The model is a discretization of the noncommutative gauge theories that arise from toroidal compactification of Matrix theory and it includes a recent proposal for a non-perturbative definition of noncommutative Yang-Mills theory in terms of twisted reduced models. The model is interpreted as a manifestly star-gauge invariant lattice formulation of noncommutative gauge theory, which reduces to ordinary Wilson lattice gauge theory for particular choices of parameters. It possesses a continuum limit which maintains both finite spacetime volume and finite noncommutativity scale. We show how the matrix model may be used for studying the properties of noncommutative gauge theory.

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arXiv1999

Making the gravitational path integral more Lorentzian, or: Life beyond Liouville gravity

In two space-time dimensions, there is a theory of Lorentzian quantum gravity which can be defined by a rigorous, non-perturbative path integral and is inequivalent to the well-known theory of (Euclidean) quantum Liouville gravity. It has a number of appealing features: i) its quantum geometry is non-fractal, ii) it remains consistent when coupled to matter, even beyond the c=1 barrier, iii) it is closer to canonical quantization approaches than previous path-integral formulations, and iv) its construction generalizes to higher dimensions.

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arXiv1999

Connected Correlators in Random Geometries

We analyze correlation functions in a toy model of a random geometry interacting with matter. We show that in general the connected correlator will contain a long-range scaling part which is in some sense a remnant of the disconnected part. This result supports the previously conjectured general form of correlation functions. We discuss the interplay between matter and geometry and the role of the symmetry in the matter sector.

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