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Conformal Field Theory of Free Bosonic and Fermionic Fields

Giuseppe Mussardo

Published Mar 26, 2020• 0 citations
Publisher

Abstract

Abstract Free theories are usually regarded as trivial examples of quantum systems. This chapter proves that this is not the case of the conformal field theories associated to the free bosonic and fermionic fields. The subject is not only full of beautiful mathematical identities but is also the source of deep physical concepts with far reaching applications. Chapter 12 also covers quantization of the bosonic field, vertex operators, the free bosonic field on a torus, modular transformations, the quantization of the free Majorana fermion, the Neveu–Schwarz and Ramond sectors, fermions on a torus, calculus for anti-commuting quantities and partition functions.

DOI
10.1093/oso/9780198788102.003.0012

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Abstract Free theories are usually regarded as trivial examples of quantum systems. This chapter proves that this is not the case of the conformal field theories associated to the free bosonic and fermionic fields. The subject is not only full of beautiful mathematical identities but is also the source of deep physical concepts with far reaching applications. Chapter 12 also covers quantization of the bosonic field, vertex operators, the free bosonic field on a torus, modular transformations, the quantization of the free Majorana fermion, the Neveu–Schwarz and Ramond sectors, fermions on a torus, calculus for anti-commuting quantities and partition functions.
Abstract In this chapter we discuss two explicit examples of conformal field theories. We start our analysis with the free massless bosonic theory that we have already seen in the previous chapter. After, we discuss the conformal field theory of a complex fermion operator (a Dirac fermion) using its decomposition in the real Majorana components. The central charge of both bosonic and fermion theories is c = 1 and this suggests the existence of an equivalence between them. The transformation that maps a bosonic into a fermion theory and vice versa is known as bosonization: it provides a useful tool both for the comprehension of the conformal theories and for a wide range of applications, in particular in low-dimensional condensed matter systems.
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}.
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.
Essential contents on quantization of fields and second quantization, which are necessary for undergraduate and graduate level, are presented in this chapter. An innovative approach is used for the presentations and the discussions on field quantization and second quantization for systems of Bosons and Fermions. The chapter covers field operators, the procedure of second quantization and its use for the Schrödinger equation for both Bosons and Fermions, matrix representation of Fermionic operators, number operators for both Bosons and Fermions, and systems of weakly interacting Bosons and Fermions. A good number of solved examples and unsolved exercises clarify the basic concepts behind quantization of fields and second quantization.
Quantum fields and quantization are concepts that come from quantum physics, the most intriguing physical theory developed in the twentieth century. In our work we would like to describe in a coherent and comprehensive way basic aspects of their mathematical structure. Most of our work is devoted to the simplest kinds of quantum fields and of quantization. We will mostly discuss mathematical aspects of free quantum fields. We will consider the quantization only on linear phase spaces. The reader will see that even within such a restricted scope the subject is rich, involves many concepts and has important applications, both to quantum theory and to pure mathematics. A distinguished role in our work will be played by representations of the canonical commutation and anti-commutation relations . Let us briefly discuss the origin and the meaning of these concepts.