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arXiv:1602.01190 · International Journal of Modern Physics A

Dipolar quantization and the infinite circumference limit of two-dimensional conformal field theories

Nobuyuki Ishibashi, Tsukasa Tada

Published Nov 14, 2016• 41 citations
arXivHTMLPDF

Abstract

Elaborating on our previous presentation, where the term dipolar quantization was introduced, we argue here that adopting [Formula: see text] as the Hamiltonian instead of [Formula: see text] yields an infinite circumference limit in two-dimensional conformal field theory. The new Hamiltonian leads to dipolar quantization instead of radial quantization. As a result, the new theory exhibits a continuous and strongly degenerated spectrum in addition to the Virasoro algebra with a continuous index. Its Hilbert space exhibits a different inner product than that obtained in the original theory. The idiosyncrasy of this particular Hamiltonian is its relation to the so-called sine-square deformation, which is found in the study of a certain class of quantum statistical systems. The appearance of the infinite circumference explains why the vacuum states of sine-square deformed systems are coincident with those of the respective closed-boundary systems.

DOI
10.1142/s0217751x16501700

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Elaborating on our previous presentation, where the term dipolar quantization was introduced, we argue here that adopting [Formula: see text] as the Hamiltonian instead of [Formula: see text] yields an infinite circumference limit in two-dimensional conformal field theory. The new Hamiltonian leads to dipolar quantization instead of radial quantization. As a result, the new theory exhibits a continuous and strongly degenerated spectrum in addition to the Virasoro algebra with a continuous index. Its Hilbert space exhibits a different inner product than that obtained in the original theory. The idiosyncrasy of this particular Hamiltonian is its relation to the so-called sine-square deformation, which is found in the study of a certain class of quantum statistical systems. The appearance of the infinite circumference explains why the vacuum states of sine-square deformed systems are coincident with those of the respective closed-boundary systems.
We argue that an infinite circumference limit can be obtained in two-dimensional conformal field theory by adopting as a Hamiltonian instead of L 0 . The theory obtained has a circumference of infinite length and hence exhibits a continuous and heavily degenerated spectrum as well as the continuous Virasoro algebra. The choice of this Hamiltonian was inspired partly by the so-called sine-square deformation, which is found in the study of a certain class of quantum statistical systems. The enigmatic behavior of sine-square deformed systems such as the sharing of their vacuum states with the closed boundary systems can be understood by the appearance of an infinite circumference.
Motivated by sine-square deformation (SSD) for quantum critical systems in 1+1 dimensions, we discuss a Möbius quantization approach to the 2D conformal field theory, which bridges the conventional radial quantization and the dipolar quantization recently proposed by Ishibashi and Tada [N. Ishibashi and T. Tada, J. Phys. A: Math. Theor. 48, 315402 (2015)], [N. Ishibashi and T. Tada, arXiv:1602.01190 [hep-th] [Search INSPIRE]. We then find that the continuous Virasoro algebra of the dipolar quantization can be interpreted as a continuum limit of the Virasoro algebra for scaled generators in the SSD limit of the Möbius quantization approach.
This article presents a systematic development of conformal field theory from first principles in quantum field theory, emphasizing complete derivations of the conformal group in arbitrary dimensions from the conformal Killing equation, proving its isomorphism to SO(d+1,1)\mathrm{SO}(d+1,1)SO(d+1,1), and developing the embedding space formalism as a tool for constructing conformally covariant correlation functions constrained by Ward identities. In two dimensions, conformal symmetry enhances to an infinite-dimensional algebra: we derive the Virasoro algebra [Lm,Ln]=(m−n)Lm+n+c12(m3−m)δm+n,0[L_m, L_n] = (m-n)L_{m+n} + \frac{c}{12}(m^3-m)\delta_{m+n,0}[Lm​,Ln​]=(m−n)Lm+n​+12c​(m3−m)δm+n,0​ from infinitesimal generators ℓn=−zn+1∂z\ell_n = -z^{n+1}\partial_zℓn​=−zn+1∂z​, computing the central charge explicitly for free theories (boson c=1c=1c=1, Majorana fermion c=1/2c=1/2c=1/2, bcbcbc-ghosts c=1−3(2λ−1)2c = 1-3(2\lambda-1)^2c=1−3(2λ−1)2) and demonstrating its physical interpretation as vacuum energy. Through radial quantization, we establish the operator-state correspondence and construct the Hilbert space via Virasoro descendants of primary states ∣h,hˉ⟩|h,\bar{h}\rangle∣h,hˉ⟩, deriving the transformation law T′(w)=(dw/dz)−2[T(z)−(c/12){w;z}]T'(w) = (dw/dz)^{-2}[T(z) - (c/12)\{w;z\}]T′(w)=(dw/dz)−2[T(z)−(c/12){w;z}] that reveals the geometric origin of the conformal anomaly.
The fundamentals of the conformal symmetry transformations in two dimensions are introduced. From the adapted \textit{Ward identities under conformal constraints}, the \textit{operator product expansion} (OPE) is defined for free fields and the energy momentum tensor (EM). After demonstrating how the energy momentum tensor transforms under conformal invariance, the notion of the \textit{central charge c} is defined. Conformal fields in two dimensions are \textit{radially} quantized on the cylinder within the framework of the second quantization of quantum field theory and are then mapped onto complex plane via a conformal mapping. The \textit{radial ordering} of conformal fields is adopted. The \textit{Virasoro algebra} is derived from the identification of the \textit{conformal generators}, which follow from relations between equal-time commutators and contour integrals of conformal fields. The radial quantization is applied to the free fermionic and bosonic fields, where for the latter the \textit{Vertex operator} is defined and specified. The Fock space of the free boson is constructed through the \textit{operator state correspondence}. Various boundary conditions are applied to free fields, and their effects on the ground-state energy density are detailed, leading to the notion of the \textit{Casimir energy}. The OPEs are regularized for the \textit{interacting} conformal fields, namely the E-M tensor, and the related definition of \textit{normal ordering} is revisited. Utilization of the Virasoro algebra led first to the \textit{descendant states} and subsequently to the \textit{conformal families}. The OPEs are generalized for the conformal families, and it is demonstrated how the three-point function coefficients may be obtained in terms of the two-point functions. The same procedure is employed generically through the definition of the \textit{conformal blocks}. The \textit{conformal Bootstrap method} is introduced to calculate the coefficients of nnn point functions. The Hilbert space of the conformal fields in two dimensions is formally constructed through the \textit{representation theory of Virasoro algebra}. Accordingly, the reducible and irreducible \textit{Verma modules} are specified. Due to the unitarity requirement of the Hilbert space, the famous \textit{Kac determinant} is introduced to obtain constraints on the inner products of the Verma modules, whence on the \textit{conformal weights} and the central charge. The decoupling of the \textit{null states} from rest of the their conformal families is used to obtain \textit{fusion rules} of the families. The \textit{Kac table} for the constrained conformal weights is constructed, and its analysis under the requirement of unitarity leads to the identification of the \textit{minimal models}. The correspondence between the discrete statistical systems at criticality and the minimal models is studied, where special attention is given on the \textit{Ising model}.
We study the properties of two-dimensional conformal field theories (CFTs) with sine-square deformation (SSD). We show that there are no eigenstates of the finite norm for the Hamiltonian of a unitary CFT with SSD, except for the zero-energy vacuum state |∣0⟩\left| {0} \right\rangle∣0⟩|⁠. We then introduce a regularized version of the SSD Hamiltonian which is related to the undeformed Hamiltonian via a unitary transformation corresponding to the Möbius quantization. The unitary equivalence of the two Hamiltonians allows us to obtain zero-energy states of the deformed Hamiltonian in a systematic way. The regularization also provides a way to compute the expectation values of observables in zero-energy states that are not necessarily normalizable.