Geometrical structures in (Q)FT

Applying recursive renormalization group transformations to a scalar field theory, we obtain an effective quantum gravity theory with an emergent extra dimension, described by a dual holographic Einstein-Klein-Gordon type action. Here, the dynamics of both the dual order-parameter field and the metric tensor field originate from density-density and energy-momentum tensor-tensor effective interactions, respectively, in the recursive renormalization group transformation, performed approximately in the Gaussian level. This linear approximation in the recursive renormalization group transformation for the gravity sector gives rise to a linearized quantum Einstein-scalar theory along the z-directional emergent space. In the large N limit, where N is the flavor number of the original scalar fields, quantum fluctuations of both dynamical metric and dual scalar fields are suppressed, leading to a classical field theory of the Einstein-scalar type in (D+1)-spacetime dimensions. We show that this emergent background gravity describes the renormalization group flows of coupling functions in the UV quantum field theory through the extra dimension. More precisely, the IR boundary conditions of the gravity equations correspond to the renormalization group β-functions of the quantum field theory, where the infinitesimal distance in the extra-dimensional space is identified with an energy scale for the renormalization group transformation. Finally, we also show that this dual holographic formulation describes quantum entanglement in a geometrical way, encoding the transfer of quantum entanglement from quantum matter to classical gravity in the large N limit. We claim that this entanglement transfer serves as a microscopic foundation for the emergent holographic duality description.

We construct a consistent quantum field theory of a free massless (pseudo)scalar field in 1+1-dimensional space-times free of infrared divergences. We show that in such a quantum field theory (i) a continuous symmetry of (pseudo)scalar field translations is spontaneously broken, (ii) Goldstone bosons appear as quanta of a free massless (pseudo)scalar field and (iii) there is a non-vanishing spontaneous {magnetization}. In spite of the existence of a spontaneous {magnetization} the main inequality between vacuum expectation values of certain operators which have been used for the derivation of the Mermin–Wagner–Hohenberg theorem (C. Itzykson and J.-M. Drouffe, {Statistical field theory}, Vol. I, 1989, pp. 219–224) is fulfilled.

A recently-proposed technique, called the dimensional expansion, uses the space-time dimension $D$ as an expansion parameter to extract nonperturbative results in quantum field theory. Here we apply dimensional-expansion methods to examine the Ising limit of a self-interacting scalar field theory. We compute the first few coefficients in the dimensional expansion for $\gamma_{2n}$, the renormalized $2n$-point Green's function at zero momentum, for $n\!=\!2$, 3, 4, and 5. Because the exact results for $\gamma_{2n}$ are known at $D\!=\!1$ we can compare the predictions of the dimensional expansion at this value of $D$. We find typical errors of less than $5\%$. The radius of convergence of the dimensional expansion for $\gamma_{2n}$ appears to be ${{2n}\over {n-1}}$. As a function of the space-time dimension $D$, $\gamma_{2n}$ appears to rise monotonically with increasing $D$ and we conjecture that it becomes infinite at $D\!=\!{{2n}\over {n-1}}$. We presume that for values of $D$ greater than this critical value, $\gamma_{2n}$ vanishes identically because the corresponding $\phi^{2n}$ scalar quantum field theory is free for $D\!>\!{{2n}\over{n-1}}$.

The geometric structure of S-matrix encapsulated by the “Amplituhedron program” has begun to reveal itself even in non-supersymmetric quantum field theories. Starting with the seminal work of Arkani-Hamed, Bai, He and Yan [1] it is now understood that for a wide class of scalar quantum field theories, tree-level amplitudes are canonical forms associated to polytopes known as accordiohedra. Similarly the higher loop scalar integrands are canonical forms associated to so called type-D cluster polytopes for cubic interactions or recently discovered class of polytopes termed pseudo-accordiohedron for higher order scalar interactions.In this paper, we continue to probe the universality of these structures for a wider class of scalar quantum field theories. More in detail, we discover new realisations of the associahedron in planar kinematic space whose canonical forms generate (colour-ordered) tree-level S matrix of external massless particles with n − 4 massless poles and one massive pole at m2. The resulting amplitudes are associated to λ1 {phi}_1^3 + λ2 {phi}_1^2 ϕ2 potential where ϕ1 and ϕ2 are massless and massive scalar fields with bi-adjoint colour indices respectively. We also show how in the “decoupling limit” (where m → ∞, λ2 → ∞ such that g : frac{uplambda_2}{m} = finite) these associahedra project onto a specific class of accordiohedron which are known to be positive geometries of amplitudes generated by λ {phi}_1^3 + g {phi}_1^4 .

Introduction This chapter develops the ideas of classical field theory in the context of special relativity. We use a scalar field and the electromagnetic field as examples of classical fields. The discussion of scalar field theory will allow us to understand concepts that are unique to field theory in a somewhat simpler context than electromagnetism; it will also be useful later on in the study of topics such as inflation, quantum field theory in curved spacetime, etc. As regards electromagnetism, we concentrate on those topics that will have direct relevance in the development of similar ideas in gravity (gauge invariance, Hamilton–Jacobi theory for particle motion, radiation and radiation reaction, etc.). The ideas developed here will be used in the next chapter to understand why a field theory of gravity – developed along similar lines – runs into difficulties. The concept of an action principle for a field will be extensively used in Chapter 6 in the context of gravity. Other topics will prove to be valuable in studying the effect of gravity on different physical systems. External fields of force In non-relativistic mechanics, the effect of an external force field on a particle can be incorporated by adding to the Lagrangian the term − V(t, x) , thereby adding to the action the integral of − V dt . Such a modification is, however, not Lorentz invariant and hence cannot be used in a relativistic theory.

Covariant, self-interacting scalar quantum field theories admit solutions for low enough spacetime dimensions, but when additional divergences appear in higher dimensions, the traditional approach leads to results, such as triviality, that are less than satisfactory. Guided by idealized but soluble nonrenormalizable models, a nontraditional proposal for the quantization of covariant scalar field theories is advanced, which achieves a term-by-term, divergence-free, perturbation analysis of interacting models expanded about a suitable pseudofree theory, which differs from a free theory by an O(ℏ2) counterterm. These positive features are realized within a functional integral formulation by a local, nonclassical, counterterm that effectively transforms parameter changes in the action from generating mutually singular measures, which are the basis for divergences, to equivalent measures, thereby removing all divergences. The use of an alternative model about which to perturb is already supported by properties of the classical theory and is allowed by the inherent ambiguity in the quantization process itself. This procedure not only provides acceptable solutions for models for which no acceptable, faithful solution currently exists, e.g. φ4n, for spacetime dimensions n ⩾ 4, but offers a new, divergence-free solution for less-singular models as well, e.g. φ4n, for n = 2, 3. Our analysis implies similar properties for multicomponent scalar models, such as those associated with the Higgs model.

In this talk we introduce a new technique, called the delta expansion, which can be used to solve nonlinear problems in both classical and quantum physics. The idea of the delta expansion is to expand in the power of a nonlinear term. For example, to treat a y4 term, we introduce a small parameter delta and consider a (y2(1+δ)) term. When we expand in powers of delta, the resulting perturbation series appears to have a finite radius of convergence and numerical results are superb. We illustrate the delta expansion by applying it to various difficult nonlinear equations taken from classical physics, the Lane-Emden, Thomas-Fermi, Blasius, Duffing, Burgers, and Kordeweg-de Vries equations. In the study of quantum field theory, the delta expansion is a powerful calculational tool that provides nonperturbative information. The basic idea is to expand in the power of the interaction term. For example, to solve a λφ4 theory in d-dimensional space-time, we introduce a small parameter δ and consider a λ(φ2)1+δ field theory. We show how to expand such a theory as a series in powers of δ using graphical methods. The resulting perturbation series appears to have a finite radius of convergence and numerical results for low-dimensional models are good. We compute the Green's functions for a scalar quantum field theory to first order in delta, renormalize the theory, and see that when the space-time dimension is four or more, the theory is free. This conclusion remains valid to second order in delta, and we believe that it remains valid to all orders in delta. The delta expansion is consistent with global supersymmetry invariance. We examine a supersymmetric quantum field theory for which we do not know of any other means for doing analytical calculations and compute the ground-state energy density and the fermion-boson mass ratio to second order in delta. Last, we show how to use the delta expansion to solve theories having a local gauge invariance. We compute the anomaly in two-dimensional electrodynamics and discuss the calculation of g-2 in four-dimensional electrodynamics.

We construct a class of topological field theories with Wess-Zumino term in spacetime dimensions ≥ 2 whose target space has a geometrical structure that suitably generalizes Poisson or twisted Poisson manifolds. Assuming a field content comprising a set of scalar fields accompanied by gauge fields of degree (1, p − 1, p) we determine a generic Wess-Zumino topological field theory in p + 1 dimensions with background data consisting of a Poisson 2-vector, a (p + 1)-vector R and a (p + 2)-form H satisfying a specific geometrical condition that defines a H-twisted R-Poisson structure of order p + 1. For this class of theories we demonstrate how a target space covariant formulation can be found by means of an auxiliary connection without torsion. Furthermore, we study admissible deformations of the generic class in special spacetime dimensions and find that they exist in dimensions 2, 3 and 4. The two-dimensional deformed field theory includes the twisted Poisson sigma model, whereas in three dimensions we find a more general structure that we call bi-twisted R-Poisson. This extends the twisted R-Poisson structure of order 3 by a non-closed 3-form and gives rise to a topological field theory whose covariant formulation requires a connection with torsion and includes a twisted Poisson sigma model in three dimensions as a special case. The relation of the corresponding structures to differential graded Q-manifolds based on the degree shifted cotangent bundle T*[p]T*[1]M is discussed, as well as the obstruction to them being QP-manifolds due to the Wess-Zumino term.

Quantum field theory in the large N limit: a review

Monte Carlo calculations in the framework of lattice field theory provide non-\n\nperturbative access to the equilibrium physics of quantum fields. When applied to\n\ncertain fermionic systems, or to the calculation of out-of-equilibrium physics, these\n\nmethods encounter the so-called sign problem, and computational resource require-\n\nments become impractically large. These difficulties prevent the calculation from\n\nfirst principles of the equation of state of quantum chromodynamics, as well as the\n\ncomputation of transport coefficients in quantum field theories, among other things.\n\nThis thesis details two methods for mitigating or avoiding the sign problem.\n\nFirst, via the complexification of the field variables and the application of Cauchy’s\n\nintegral theorem, the difficulty of the sign problem can be changed. This requires\n\nsearching for a suitable contour of integration. Several methods of finding such a\n\ncontour are discussed, as well as the procedure for integrating on it. Two notable\n\nexamples are highlighted: in one case, a contour exists which entirely removes the\n\nsign problem, and in another, there is provably no contour available to improve the sign problem by more than a (parametrically) small amount.\n\nAs an alternative, physical simulations can be performed with the aid of a\n\nquantum computer. The formal elements underlying a quantum computation —\n\nthat is, a Hilbert space, unitary operators acting on it, and Hermitian observables\n\nto be measured — can be matched to those of a quantum field theory. In this way\n\nan error-corrected quantum computer may be made to serve as a well controlled\n\nlaboratory. Precise algorithms for this task are presented, specifically in the context\n\nof quantum chromodynamics.

Providing a new perspective on quantum field theory, this book gives a pedagogical and up-to-date exposition of non-perturbative methods in relativistic quantum field theory and introduces the reader to modern research work in theoretical physics. It describes in detail non-perturbative methods in quantum field theory, and explores two- dimensional and four- dimensional gauge dynamics using those methods. The book concludes with a summary emphasizing the interplay between two- and four- dimensional gauge theories. Aimed at graduate students and researchers, this book covers topics from two-dimensional conformal symmetry, affine Lie algebras, solitons, integrable models, bosonization, and 't Hooft model, to four-dimensional conformal invariance, integrability, large N expansion, Skyrme model, monopoles and instantons. Applications, first to simple field theories and gauge dynamics in two dimensions, and then to gauge theories in four dimensions and quantum chromodynamics (QCD) in particular, are thoroughly described.

Effective field theories (EFTs) are used whenever one encounters problems with two disparate scales, a high-energy scale Λ h and a lower scale Λ l , in quantum field theory. EFTs allow one to expand physical quantities in the small ratio of the scales and to separate the low-energy contributions from the high-energy part. Performing the expansion usually greatly simplifies the problem and is often necessary in order to be able to attack a field-theory problem in the first place. In Quantum Chromodynamics (QCD), the low-energy part is usually non-perturbative, while the high-energy contribution can be computed perturbatively.

Of the three organizers of the conference N. Hitchin is a mathematician, A. Kapustin and W. Nahm are physicists. Ideas from physics, in particular from string theory and related areas had a profound influence on algebraic and differential geometry. Conversely, many developments in string theory use very recent mathematical results. There were 19 talks at the workshop, 12 by mathematicians and 7 by physicists, but in many cases an outside observer would have had difficulties to sort them out. Quantum field theory often allows continuous interpolations between geometrical structures of topologically different manifolds. A particularly important example is mirror symmetry, where the complexified Kähler cone of one manifold corresponds to the moduli space of complex structures of the mirror manifold. Several talks explored geometrical and number theoretical aspects of this connection. Mirror symmetry can be understood in terms of torus fibrations and an isomorphism between the quantum field theories given by maps to a torus and to its dual (T-duality). The B-field of string theory can be described in terms of non-commutative geometry on the tori, and its effect on T-duality has been investigated. The quantum field theories which provide these dualities are supersymmetric conformally invariant quantum field theories in two dimensions. In mathematics, large parts of their structures have been axiomatised in the language of vertex operator algebras, but much remains to be done. A promising mathematical approach is provided by the chiral de Rham complex. This cohomology of this complex seems to be invariant under mirror symmetry, but its physical meaning is not yet clear. This led to many discussions between mathematicians and physicists. Indeed, the free time for discussions was at least as important as the lectures for learning the language of researchers with a different background and for making use of their ideas. An area in which the common understanding is well advance is topological field theory. Here one has good axioms and many of the analytic problems of conformally invariant quantum field theories can be ignored. One can solve rather complex problems and one can make contact with new geometrical ideas, in particular generalised complex structures. The latter allow to interpolate between complex and symplectic geometry, which should become important for both physics and mathematics. To understand all of conformal field theory at a similar mathematical depth will take more time, but already now relations to mixed Hodge structures have given much insight. Eventually many so far intractable problems should be solvable by the use of quantum field theoretical ideas, for example the construction of the non-singular Ricci flat metrics on K3. Perturbative string theory can be described in terms of conformal field theory, but the complete theory has more excitations than strings. Among them the best understood are the D-branes, many features of which can be understood in terms of boundary states in conformal field theory. Some of these branes are related to complex submanifolds, others to certain Lagrangian submanifolds, and their description involves twisted K-theory and derived categories. The development of a common language between mathematicians and physicists is well under way, and the workshop has made a contribution which all participants found very stimulating. In Europe such meetings between mathematicians and string theorists are rare, so the workshop was particularly useful. Much of the time was reserved for informal discussions. On one evening there was a discussion on the relation between classical and quantum geometry. Weather predictions for the free afternoon had been bad, but finally the weather was splendid and the supply of Black Forest cake matched the demand.

Recent arguments have indicated that the sum over histories formulation of quantum amplitudes for gravity should include sums over conifolds, a set of histories with more general topology than that of manifolds. This paper addresses the consequences of conifold histories in gravitational functional integrals that also include scalar fields. This study will be carried out explicitly for the generalized Hartle-Hawking initial state, that is the Hartle-Hawking initial state generalized to a sum over conifolds. In the perturbative limit of the semiclassical approximation to the generalized Hartle-Hawking state, one finds that quantum field theory on Einstein conifolds is recovered. In particular, the quantum field theory of a scalar field on de Sitter spacetime with ${\mathrm{RP}}^{3}$ spatial topology is derived from the generalized Hartle-Hawking initial state in this approximation. This derivation is carried out for a scalar field of arbitrary mass and scalar curvature coupling. Additionally, the generalized Hartle-Hawking boundary condition produces a state that is not identical to but corresponds to the Bunch-Davies vacuum on ${\mathrm{RP}}^{3}$ de Sitter spacetime. This result cannot be obtained from the original Hartle-Hawking state formulated as a sum over manifolds as there is no Einstein manifold with round ${\mathrm{RP}}^{3}$ boundary.

The black hole information loss paradox epitomizes the contradictions between general relativity and quantum field theory. The AdS/conformal field theory (CFT) correspondence provides an implicit answer for the information loss paradox in black hole physics by equating a gravity theory with an explicitly unitary field theory. Gravitational collapse in asymptotically AdS spacetimes is generically turbulent. Given that the mechanism to read out the information about correlations functions in the field theory side is plagued by deterministic classical chaos, we argue that quantum chaos might provide the true Rosetta Stone for answering the information paradox in the context of the AdS/CFT correspondence.