This paper is mainly based on the talk I presented at the meeting "The Philosophy and Physics of Noether's Theorems" that took place 5-6 October 2018, but it also contains some original results that were inspired by discussions with mathematicians, physicists and philosophers about the problem of understanding the intrinsic meaning of gauge invariance. In this work, I argue that following the principles of locality, deformation and homology, one naturally ends up using the Batalin-Vilkovisky (BV) formalism in quantizing gauge theories.
I start with the gentle introduction into the BV framework and then I proceed to some new results and more speculative deliberations. In the classical theory, I present a new perspective on the classical BV operator, using the notion of Moller maps. In the quantum theory, I present some loose ideas on the formulation of anomalous master Ward identity in the framework proposed recently by Buchholz and Fredenhagen, based on local S-matrices.
We present a general construction of KMS states in the framework of perturbative algebraic quantum field theory (pAQFT). Our approach may be understood as an extension of the Schwinger-Keldysh formalism. We obtain in particular the Wightman functions at positive temperature, thus solving a problem posed some time ago by Steinmann. The notorious infrared divergences observed in a diagrammatic expansion are shown to be absent due to a consequent exploitation of the locality properties of pAQFT. To this avail, we introduce a novel, Hamiltonian description of the interacting dynamics and find, in …
We present the first rigorous construction of the QFT Thirring model, for any value of the mass, in a functional integral approach, by proving that a set of Grassmann integrals converges, as the cutoffs are removed, to a set of Schwinger functions verifying the Osterwalder-Schrader axioms. The massless limit is investigated and it is shown that the Schwinger functions have different properties with respect to the ones of the well known exact solution: the Ward Identities have anomalies violating the anomaly non-renormalization property and additional anomalies, apparently unnoticed before, are…
We analyze the asymptotically free massless scalar $ϕ^3$ quantum field theory in 6 dimensions, using resurgent asymptotic analysis to find the trans-series solutions which yield the non-perturbative completion of the divergent perturbative solutions to the Kreimer-Connes Hopf-algebraic Dyson-Schwinger equations for the anomalous dimension. This scalar conformal field theory is asymptotically free and has a real Lipatov instanton. In the Hopf-algebraic approach we find a trans-series having an intricate Borel singularity structure, with three distinct but resonant non-perturbative terms, each r…
Exact non-perturbative partition functions of coupling constants and external fields exhibit huge hidden symmetry, reflecting the possibility to change integration variables in the functional integral. In many cases this implies also some non-linear relations between correlation functions, typical for the tau-functions of integrable systems. To a variety of old examples, from matrix models to Seiberg-Witten theory and AdS/CFT correspondence, now adds the Chern-Simons theory of knot invariants. Some knot polynomials are already shown to combine into tau-functions, the search for entire set of r…
We will present the axioms of Bogoliubov's causal perturbative QFT in which the creation-annihilation operators are interpreted as Hida operators. We will shortly present the results that can be achieved in this theory: 1. Removal of UV and IR infinity in the scattering operator, 2. Existence of the adiabatic limit for interacting fields in QED, 3. Proof that charged particles have non-zero mass, 4. Existence of infrared and ultraviolet asymptotics for QED.
We study the perturbative quantization of 2-dimensional massive scalar field theory with polynomial (or power series) potential on manifolds with boundary. We prove that it fits into the functorial quantum field theory framework of Atiyah-Segal. In particular, we prove that the perturbative partition function defined in terms of integrals over configuration spaces of points on the surface satisfies an Atiyah-Segal type gluing formula. Tadpoles (short loops) behave nontrivially under gluing and play a crucial role in the result.
For any near-threshold asymptotic regime and for any Feynman diagram (involving loop and/or phase space integrals), a systematic prescription for explicitly constructing all-logs, all-powers (all-twists) expansions in perfectly factorized form with explicit integrals for coefficients, is presented. The distribution-theoretic nature of the method of asymptotic operation employed allows treatment of totally exclusive phase space in the same manner as loop integrals.
Recent years have seen noteworthy progress in the mathematical formulation of quantum field theory and perturbative string theory. We give a brief survey of these developments. It serves as an introduction to the more detailed collection "Mathematical Foundations of Quantum Field Theory and Perturbative String Theory".
It is well-known that there exist infinitely-many inequivalent representations of the canonical (anti)-commutation relations of Quantum Field Theory (QFT). A way out, suggested by Algebraic QFT, is to instead define the quantum theory as encompassing all possible (abstract) states. In the present paper, we describe a quantization scheme for general linear (aka. free) field theories that can be seen as intermediate between traditional Fock quantization and full Algebraic QFT, in the sense that: * it provides a constructive, explicit description of the resulting space of quantum states; * it doe…
This is the first in a series of papers on an attempt to understand quantum field theory mathematically. In this paper we shall introduce and study BV QFT algebra and BV QFT as the proto-algebraic model of quantum field theory by exploiting Batalin-Vilkovisky quantization scheme. We shall develop a complete theory of obstruction (anomaly) to quantization of classical observables and propose that expectation value of quantized observable is certain quantum homotopy invariant. We shall, then, suggest a new method, bypassing Feynman's path integrals, of computing quantum correlation functions whe…
Symmetry Breaking is used as an "underlying principle", bringing different features of QFT to the foreground. However, the understanding of Symmetry Breaking that is used here is quite different from what is done in the mainstream: Symmetry Breaking is understood as the solution set of a given QFT, its vacuum manifold, or, more modernly, its Moduli Space. Distinct solutions correspond to different sectors, phases, of the theory, which are nothing but distinct foliations of the vacuum manifold, or points in the Moduli Space (for all possible values of the parameters of the theory). Under this f…
The structure of overlapping subdivergences, which appear in the perturbative expansions of quantum field theory, is analyzed using algebraic lattice theory. It is shown that for specific QFTs the sets of subdivergences of Feynman diagrams form algebraic lattices. This class of QFTs includes the Standard model. In kinematic renormalization schemes, in which tadpole diagrams vanish, these lattices are semimodular. This implies that the Hopf algebra of Feynman diagrams is graded by the coradical degree or equivalently that every maximal forest has the same length in the scope of BPHZ renormaliza…
We present several approaches to renormalization in QFT: the multi-scale analysis in perturbative renormalization, the functional methods à la Wetterich equation, and the loop-vertex expansion in non-perturbative renormalization. While each of these is quite well-established, they go beyond standard QFT textbook material, and may be little-known to specialists of each other approach. This review is aimed at bridging this gap.
The present state of QFT is analysed from a new viewpoint whose mathematical basis is the modular theory of von Neumann algebras. Its physical consequences suggest new ways of dealing with interactions, symmetries, Hawking-Unruh thermal properties and possibly also extensions of the scheme of renormalized perturbation theory. Interactions are incorporated by using the fact that the S-matrix is a relative modular invariant of the interacting- relative to the incoming- net of wedge algebras. This new point of view allows many interesting comparisions with the standard quantization approach to QF…
We introduce a non-perturbative continuum framework to study the dynamics of quantum field theory (QFT), applied here to the CP(N-1) model, using Ecalle's theory of resurgent trans-series, combined with the physical principle of continuity, in which spatial compactification and a Born-Oppenheimer approximation reduce QFT to quantum mechanics, while preventing all intervening rapid cross-overs or phase transitions. The reduced quantum mechanics contains the germ of all non-perturbative data, e.g., mass gap, of the QFT, all of which are calculable. For CP(N-1), the results obtained at arbitrary …
Although perturbative quantum field theory is highly successful, it possesses a number of well-known analytic problems, from ultraviolet and infrared divergencies to the divergence of the perturbative expansion itself. As a consequence, it has been difficult, for example, to prove with full rigor that the Legendre transform of the quantum effective action is the generating functional of connected graphs. Here, we give a rigorous proof of this central fact. To this end, we show that the Legendre transform can be re-defined purely combinatorially and that it ultimately reduces to a simple homolo…
We give some remarks on exact quantization conditions associated with quantized mirror curves of local Calabi-Yau threefolds, conjectured in arXiv:1410.3382. It is shown that they characterize a non-perturbative completion of the refined topological strings in the Nekrasov-Shatashvili limit. We find that the quantization conditions enjoy an exact S-dual invariance. We also discuss Borel summability of the semi-classical spectrum.
Perturbative algebraic quantum field theory (pAQFT) is a mathematically rigorous framework that allows to construct models of quantum field theories on a general class of Lorentzian manifolds. Recently this idea has been applied also to perturbative quantum gravity, treated as an effective theory. The difficulty was to find the right notion of observables that would in an appropriate sense be diffeomorphism invariant. In this article I will outline a general framework that allows to quantize theories with local symmetries (this includes infinitesimal diffeomorphism transformations) with the us…
In this thesis, we describe some recent results obtained in the analysis of two-dimensional quantum field theories by means of semiclassical techniques. These achievements represent a natural development of the non-perturbative studies performed in the past years for conformally invariant and integrable theories, which have led to analytical predictions for several measurable quantities in the universality classes of statistical systems. Here we propose a semiclassical method to control analytically the spectrum and the finite-size effects in both integrable and non-integrable theories. The te…
We study a class of elliptic SPDEs with additive Gaussian noise on $\mathbb{R}^2 \times M$, with $M$ a $d$-dimensional manifold equipped with a positive Radon measure, and a real-valued non linearity given by the derivative of a smooth potential $V$, convex at infinity and growing at most exponentially. For quite general coefficients and a suitable regularity of the noise we obtain, via the dimensional reduction principle discussed in our previous paper on the topic, the identity between the law of the solution to the SPDE evaluated at the origin with a Gibbs type measure on the abstract Wiene…
These notes are based on the course given by Klaus Fredenhagen at the Les Houches Winter School in Mathematical Physics (January 29 - February 3, 2012) and the course "QFT for mathematicians" given by Katarzyna Rejzner in Hamburg for the Research Training Group 1670 (February 6 -11, 2012). Both courses were meant as an introduction to modern approach to perturbative quantum field theory and are aimed both at mathematicians and physicists.
A proposed definition is given for the quantization of a Poisson algebra, taking the quantum product to be a geodesic on the manifold of associative products.
We compute characters of the BMS group in three dimensions. The approach is the same as that performed by Witten in the case of coadjoint orbits of the Virasoro group in the eighties, within the large central charge approximation. The procedure involves finding a Poisson bracket between classical variables and the corresponding commutator of observables in a Hilbert space, explaining why we call this a quantization. We provide first a pedagogical warm up by applying the method to both SL(2,R) and Poincaré3 groups. As for BMS3, our results coincide with the characters of induced representations…
The Ward identities are the relations which the complete Green functions of quantum fields satisfy if an original classical Lagrangian system is degenerate. A generic degenerate Lagrangian system of even and odd fields is considered. It is characterized by a hierarchy of reducible Noether identities and gauge supersymmetries parameterized by antifields and ghosts, respectively. In the framework of the BV quantization procedure, an original degenerate Lagrangian is extended to ghosts and antifields in order to satisfy the master equation. Replacing antifields with gauge fixing terms, one comes …
Instead of formulating the states of a Quantum Field Theory (QFT) as density matrices over a single large Hilbert space, it has been proposed by Kijowski [Kijowski, 1977] to construct them as consistent families of partial density matrices, the latter being defined over small 'building block' Hilbert spaces. In this picture, each small Hilbert space can be physically interpreted as extracting from the full theory specific degrees of freedom. This allows to reduce the quantization of a classical field theory to the quantization of finite-dimensional sub-systems, thus sidestepping some of the co…
We report a rigorous quantization of topological quantum mechanics on $\mathbb{R}_{\geqslant 0}$ and $\mathbf{I}= [0, 1]$ in perturbative BV-BFV formalism. Costello's homotopic renormalization is extended, and incorporated in our construction. Moreover, BV quantization of the same model studied in our previous work is derived from the BV-BFV quantization, leading to a comparison between different interpretations of ''state'' in these two frameworks.
Inspired by recent connections between spectral theory and topological string theory, we propose exact quantization conditions for the relativistic Toda lattice of N particles. These conditions involve the Nekrasov-Shatashvili free energy, which resums the perturbative WKB expansion, but they require in addition a non-perturbative contribution, which is related to the perturbative result by an S-duality transformation of the Planck constant. We test the quantization conditions against explicit calculations of the spectrum for N=3. Our proposal can be generalized to arbitrary toric Calabi-Yau m…
We demonstrate the extraordinary modernity of the 1924/25 "Einstein-Jordan fluctuation conundrum", a Gedankenexperiment which led Jordan to his quantization of waves published as a separate section in the famous Born-Heisenberg-Jordan 1926 "Dreimännerarbeit". The thermal nature of energy fluctuations caused by the restriction of the QFT vacuum to a subvolume remained unnoticed mainly because it is not present in QM. In order to understand the analogy with Einstein's fluctuation calculation in a thermal black body system, it is important to expose the mechanism which causes a global vacuum stat…
Fedosov has described a geometro-algebraic method to construct in a canonical way a deformation of the Poisson algebra associated with a finite-dimensional symplectic manifold ("phase space"). His algorithm gives a non-commutative, but associative, product (a so-called "star-product") between smooth phase space functions parameterized by Planck's constant $\hbar$, which is treated as a deformation parameter. In the limit as $\hbar$ goes to zero, the star product commutator goes to $\hbar$ times the Poisson bracket, so in this sense his method provides a quantization of the algebra of classical…
We describe a covariant framework to construct a globalized version for the perturbative quantization of nonlinear split AKSZ Sigma Models on manifolds with and without boundary, and show that it captures the change of the quantum state as one changes the constant map around which one perturbs. This is done by using concepts of formal geometry. Moreover, we show that the globalized quantum state can be interpreted as a closed section with respect to an operator that squares to zero. This condition is a generalization of the modified Quantum Master Equation as in the BV-BFV formalism, which we …
This paper contains three results about generating functions for Lie-theoretic integration of Poisson brackets and their relation to quantization. In the first, we show how to construct a generating function associated to the germ of any local symplectic groupoid and we provide an explicit (smooth, non-formal) universal formula $S_π$ for integrating any Poisson structure $π$ on a coordinate space. The second result involves the relation to semiclassical quantization. We show that the formal Taylor expansion of $S_{tπ}$ around $t=0$ yields an extract of Kontsevich's star product formula based o…
