Ivan Todorov

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Papers by Ivan Todorov

Conformal invariance in quantum field theory

La Rivista del Nuovo Cimento, 1973

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Characters and partition function for the Wen-Wu CFT model of the Haldane-Rezayi quantum Hall state

Arxiv preprint hep-th/9611084, 1996

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RCFT extensions of W 1 + ∞ in terms of bilocal fields

The rational conformal field theory (RCFT) extensions of W 1+∞ at c = 1 are in one-to-one corresp... more The rational conformal field theory (RCFT) extensions of W 1+∞ at c = 1 are in one-to-one correspondence with 1-dimensional integral lattices L(m). Each extension is associated with a pair of oppositely charged " vertex operators " of charge square m ∈ N. Their product defines a bilocal field V m (z 1 , z 2) whose expansion in powers of z 12 = z 1 − z 2 gives rise to a series of (neutral) local quasiprimary fields V l (z, m) (of dimension l + 1). The associated bilocal exponential of a normalized current generates the W 1+∞ algebra spanned by the V l (z, 1) (and the unit operator). The extension of this construction to higher (integer) values of the central charge c is also considered. Applications to a quantum Hall system require computing characters (i.e., chiral partition functions) depending not just on the modular parameter τ , but also on a chemical potential ζ. We compute such a ζ dependence of orbifold characters, thus extending the range of applications of a recen...

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Higher Dimensional Vertex Algebras and Rational Conformal Field Theory Models

The notion of global conformal invariance (GCI) in Minkowski space allows to prove rationality of... more The notion of global conformal invariance (GCI) in Minkowski space allows to prove rationality of correlation functions and to extend the concept of vertex algebra to any number D of space-time dimensions. The case of even D, which includes a conformal stress-energy tensor with a rational 3-point function, is of particular interest. Recent progress, reviewed in the talk, includes a full account of Wightman positivity at the 4-point level for D=4, and a study of modular properties of thermal expectation values of the conformal energy operator.

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Quantization is a mystery

Bulg J Phys, Jun 14, 2012

Expository notes which combine a historical survey of the development of quantum physics with a r... more Expository notes which combine a historical survey of the development of quantum physics with a review of selected mathematical topics in quantization theory (addressed to students that are not complete novices in quantum mechanics). After recalling in the introduction the early stages of the quantum revolution, and recapitulating in Sect. 2.1 some basic notions of symplectic geometry, we survey in Sect. 2.2 the so called prequantization thus preparing the ground for an outline of geometric quantization (Sect. 2.3). In Sect. 3 we apply the general theory to the study of basic examples of quantization of Kähler manifolds. In Sect. 4 we review the Weyl and Wigner maps and the work of Groenewold and Moyal that laid the foundations of quantum mechanics in phase space, ending with a brief survey of the modern development of deformation quantization. Sect. 5 provides a review of second quantization and its mathematical interpretation. We point out that the treatment of (nonrelativistic) bound states requires going beyond the neat mathematical formalization of the concept of second quantization. An appendix is devoted to Pascual Jordan, the least known among the creators of quantum mechanics and the chief architect of the "theory of quantized matter waves".

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Studying Quantum Field Theory

The paper puts together some loosely connected observations, old and new, on the concept of a qua... more The paper puts together some loosely connected observations, old and new, on the concept of a quantum field and on the properties of Feynman amplitudes. We recall, in particular, the role of (exceptional) elementary induced representations of the quantum mechanical conformal group SU(2,2) in the study of gauge fields and their higher spin generalization. A recent revival of the (Bogolubov-)Epstein-Glaser approach to position space renormalization is reviewed including an application to the calculation of residues of primitively divergent graphs. We end up with an optimistic outlook of current developments of analytic methods in perturbative QFT which combine the efforts of theoretical physicists, algebraic geometers and number theorists.

format_quoteParticles in QFT are classified by irreducible representations of the Poincaré group, indicating a long-standing tension between field and particle descriptions.format_quote

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Generalized Homologies for the Zero Modes of the SU(2) WZNW Model

Letters in Mathematical Physics, 1999

We generalize the BRS method for the (finite-dimensional) quantum gauge theory involved in the ze... more We generalize the BRS method for the (finite-dimensional) quantum gauge theory involved in the zero modes of the monodromy extended SU(2) WZNW model. The generalization consists of a nilpotent operator Q such that Qh = 0 (h = k + 2 = 2, 3, ... being the height of the current algebra representation) acting on an extended state space. The

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Euclidean Configuration Space Renormalization, Residues and Dilation Anomaly

Lie Theory and Its Applications in Physics, 2013

Configuration (x-)space renormalization of euclidean Feynman amplitudes in a massless quantum fie... more Configuration (x-)space renormalization of euclidean Feynman amplitudes in a massless quantum field theory is reduced to the study of local extensions of associate homogeneous distributions. Primitively divergent graphs are renormalized, in particular, by subtracting the residue of an analytically regularized expression. Examples are given of computing residues that involve zeta values. The renormalized Green functions are again associate homogeneous distributions of the same degree that transform under indecomposable representations of the dilation group.

format_quoteIf indices of divergence of a connected graph are negative, the corresponding Green function is locally integrable, impacting renormalization theory.format_quote

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Renormalization of massless Feynman amplitudes in configuration space

Reviews in Mathematical Physics, 2014

A systematic study of recursive renormalization of Feynman amplitudes is carried out both in Eucl... more A systematic study of recursive renormalization of Feynman amplitudes is carried out both in Euclidean and in Minkowski configuration spaces. For a massless quantum field theory (QFT), we use the technique of extending associate homogeneous distributions to complete the renormalization recursion. A homogeneous (Poincaré covariant) amplitude is said to be convergent if it admits a (unique covariant) extension as a homogeneous distribution. For any amplitude without subdivergences — i.e. for a Feynman distribution that is homogeneous off the full (small) diagonal — we define a renormalization invariant residue. Its vanishing is a necessary and sufficient condition for the convergence of such an amplitude. It extends to arbitrary — not necessarily primitively divergent — Feynman amplitudes. This notion of convergence is finer than the usual power counting criterion and includes cancellation of divergences.

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Scale and conformal covariance in quantum electrodynamics

Reports on Mathematical Physics, 1976

The paper consists of two independent parts. First, we review the situation in scale invariant ma... more The paper consists of two independent parts. First, we review the situation in scale invariant massless QED from an axiomatic standpoint. Assuming that the t-functions (or, equivalently, the Euclidean Schwinger functions) transform covariantly under dilatations, we deduce that the current &(x) has zero n-point Wightman functions, but nonvanishing t-functions. Assuming in addition conformal invariance of the current-field vertex function, we write down a bootstrap equation similar to the one derived in [7] from the point of view of perturbation theory. Next we consider a non-Lagrangian, conformal invariant model of interacting antisymmetric tensor field Fpv (of scale dimension d) and Dirac field y (of dimension d'). The model involves two conserved currents (an "electric" and a "magnetic" one) and two effective coupling constants. We demonstrate that it is free of ultraviolet divergences in the range of dimensions 2 < d < 3, 312 < d' < 512. Contents B. Scale and y5-invariance of the massless QED. The gauge invariant 2-point functions C. Gauge dependent f-point functions D. Remarks on conformal covariance and bootstrap equations. 2. A conformal covariant model of "electromagnetic" interaction with two conserved currents A. Maxwell stress tensor with anomalous dimensions. 2 and 3-point functions. Electric and magnetic currents B. Feynman rules for skeleton diagrams. Absence of ultraviolet divergences.

format_quoteReconciling conformal covariance with massless QED necessitates non-local gauges, a significant theoretical innovation.format_quote

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Partial wave expansion and Wightman positivity in conformal field theory

Nuclear Physics B, 2005

A new method for computing exact conformal partial wave expansions is developed and applied to ap... more A new method for computing exact conformal partial wave expansions is developed and applied to approach the problem of Hilbert space (Wightman) positivity in a non-perturbative four-dimensional quantum field theory model. The model is based on the assumption of global conformal invariance on compactified Minkowski space (GCI). Bilocal fields arising in the harmonic decomposition of the operator product expansion (OPE) prove to be a powerful instrument in exploring the field content. In particular, in the theory of a field L of dimension 4 which has the properties of a (gauge invariant) Lagrangian, the scalar field contribution to the 6-point function of the twist 2 bilocal field is analyzed with the aim to separate the free field part from the nontrivial part.

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Quasiprimary composite fields and operator product expansions in two-dimensional conformal models

Letters in Mathematical Physics, 1986

Renormalized composite conformal fields are constructed and applied to derive global operator pro... more

Rational conformal field theory extensions of W[sub 1+∞] in terms of bilocal fields

Journal of Mathematical Physics, 1998

The rational conformal field theory (RCFT) extensions of W 1+∞ at c = 1 are in one-to-one corresp... more The rational conformal field theory (RCFT) extensions of W 1+∞ at c = 1 are in one-to-one correspondence with 1-dimensional integral lattices L(m). Each extension is associated with a pair of oppositely charged "vertex operators" of charge square m ∈ N. Their product defines a bilocal field V m (z 1 , z 2) whose expansion in powers of z 12 = z 1 − z 2 gives rise to a series of (neutral) local quasiprimary fields V l (z, m) (of dimension l + 1). The associated bilocal exponential of a normalized current generates the W 1+∞ algebra spanned by the V l (z, 1) (and the unit operator). The extension of this construction to higher (integer) values of the central charge c is also considered. Applications to a quantum Hall system require computing characters (i.e., chiral partition functions) depending not just on the modular parameter τ , but also on a chemical potential ζ. We compute such a ζ dependence of orbifold characters, thus extending the range of applications of a recent study of affine orbifolds.

format_quoteOnly for m = 1 do the fields V l close an infinite dimensional Lie algebra, significant for describing the level c W 1+∞ structure.format_quote

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Conformal field theories with infinitely many conservation laws

Journal of Mathematical Physics, 2013

Globally conformal invariant quantum field theories in a D-dimensional space-time (D even) have r... more Globally conformal invariant quantum field theories in a D-dimensional space-time (D even) have rational correlation functions and admit an infinite number of conserved (symmetric traceless) tensor currents. In a theory of a scalar field of dimension D-2 they were demonstrated to be generated by bilocal normal products of free massless scalar fields with an O(N), U(N), or Sp(2N) (global) gauge symmetry [BNRT]. Recently, conformal field theories "with higher spin symmetry" were considered for D=3 in [MZ] where a similar result was obtained (exploiting earlier study of CFT correlators). We suggest that the proper generalization of the notion of a 2D chiral algebra to arbitrary (even or odd) dimension is precisely a CFT with an infinite series of conserved currents. We shall recast and complement (part of) the argument of Maldacena and Zhiboedov into the framework of our earlier work. We extend to D = 4 the auxiliary Weyl-spinor formalism developed in [GPY] for D = 3. The free field construction only follows for D > 3 under additional assumptions about the operator product algebra. In particular, the problem of whether a rational CFT in 4D Minkowski space is necessarily trivial remains open.

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Exchange relations and correlation functions for a quantum particle on the SU2 group manifold

Journal of Mathematical Physics, 1994

We consider the SO4 invariant quantum dynamics of a point particle moving on the 3-sphere (or equ... more We consider the SO4 invariant quantum dynamics of a point particle moving on the 3-sphere (or equivalently, of the relative motion of the spherical top). Quantum exchange relations for different times are derived with an "R matrix" depending on the time difference and on the conserved angular momentum. Their implications for correlation functions are spelled out. The chiral exchange relations of Alekseev and Faddeev [1] are also extended to different times.

format_quoteThe derived generalized exchange relations obtain correlation functions for point particles on the SU(2) manifold at varied times.format_quote

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Harmonic Bilocal Fields Generated by Globally Conformal Invariant Scalar Fields

Communications in Mathematical Physics, 2008

The twist two contribution in the operator product expansion of φ 1 (x 1) φ 2 (x 2) for a pair of... more The twist two contribution in the operator product expansion of φ 1 (x 1) φ 2 (x 2) for a pair of globally conformal invariant, scalar fields of equal scaling dimension d in four space-time dimensions is a field V 1 (x 1 , x 2) which is harmonic in both variables. It is demonstrated that the Huygens bilocality of V 1 can be equivalently characterized by a "single-pole property" concerning the pole structure of the (rational) correlation functions involving the product φ 1 (x 1) φ 2 (x 2). This property is established for the dimension d = 2 of φ 1 , φ 2. As an application we prove that any GCI scalar field of conformal dimension 2 (in four space-time dimensions) can be written as a (possibly infinite) superposition of products of free massless fields.

format_quoteV1 exists as a Huygens bilocal field for scalar fields of dimension d=2; existence not guaranteed for higher dimensions.format_quote

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Rationality of Conformally Invariant Local Correlation Functions on Compactified Minkowski Space

Communications in Mathematical Physics, 2001

Rationality of the Wightman functions is proven to follow from energy positivity, locality and a ... more Rationality of the Wightman functions is proven to follow from energy positivity, locality and a natural condition of global conformal invariance (GCI) in any number D of space-time dimensions. The GCI condition allows to treat correlation functions as generalized sections of a vector bundle over the compactification M of Minkowski space M and yields a strong form of locality valid for all non-isotropic intervals if assumed true for space-like separations.

format_quoteThe findings offer a framework for constructing conformal invariant correlation functions, critical for further QFT developments.format_quote

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Global conformal invariance and bilocal fields with rational correlation functions

Arxiv preprint hep-th/0211106, 2002

The singular part of the operator product expansion (OPE) of a pair of globally conformal invaria... more The singular part of the operator product expansion (OPE) of a pair of globally conformal invariant (GCI) scalar fields φ of (integer) dimension d can be written as a sum of the 2-point function of φ and d− 1 bilocal conformal fields V ν (x 1 , x 2) of dimension (ν, ν) , ν = 1,. .. , d − 1. As the correlation functions of φ (x) are proven to be rational [7], we argue that the correlation functions of V ν can also be assumed rational. Each V ν (x 1 , x 2) is expanded into local symmetric tensor fields of twist (dimension minus rank) 2ν. The case d = 2 , considered previously [6], is briefly reviewed and current work on the d = 4 case (of a Lagrangean density in 4 space-time dimensions) is previewed.

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Lectures on elliptic functions and modular forms in conformal field theory

Arxiv preprint math-ph/0412039, 2004

A concise review of the notions of elliptic functions, modular forms, and ϑ-functions is provided... more A concise review of the notions of elliptic functions, modular forms, and ϑ-functions is provided, devoting most of the paper to applications to Conformal Field Theory (CFT), introduced within the axiomatic framework of quantum field theory. Many features, believed to be peculiar to chiral 2D (= two dimensional) CFT, are shown to have a counterpart in any (even dimensional) globally conformal invariant quantum field theory. The treatment is based on a recently introduced higher dimensional extension of the concept of vertex algebra.

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Conformal gauges and renormalized equations of motion in massless quantum electrodynamics

Communications in Mathematical Physics, 1985

A formulation of massless QED is studied with a non-singular Lagrangian and conformal invariant e... more A formulation of massless QED is studied with a non-singular Lagrangian and conformal invariant equations of motion. It makes use of nondecomposable representations of the conformal group G and involves two dimensionless scalar fields (in addition to the conventional charged field and electromagnetic potential) but gauge invariant Green functions are shown to coincide with those of standard (massless) QED. Assuming that the (nonelementary) representation of G for the 5-potential which leaves the equations of motion invariant and leads to the free photon propagator of Johnson-Baker-Adler (JBA) conformal QED remains unaltered by renormalization, we prove that consistency requirements for conformal invariant 2-, 3-, and 4-point Green functions satisfying (renormalized) equations of motion and standard Ward identities lead to either a trivial solution (with eψ = 0) or to a subcanonical dimension d=% for the charged field.

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