Elsevier

Physics Letters B

Volume 504, Issues 1–2, 5 April 2001, Pages 109-116
Physics Letters B

Nonlinear realization of Lorentz symmetry

https://doi.org/10.1016/S0370-2693(01)00253-2Get rights and content

Abstract

We explore a nonlinear realization of the (2+1)-dimensional Lorentz symmetry with a constant vacuum expectation value of the second rank anti-symmetric tensor field. By means of the nonlinear realization, we obtain the low-energy effective action of the Nambu–Goldstone bosons for the spontaneous Lorentz symmetry breaking.

Introduction

Field theories on the space–time with non-commutative coordinates (non-commutative field theories) have been extensively studied for a few years [1], [2], [3]. A construction of the non-commutative field theories has been developed in [3]: the world volume theory of Dp-brane with a constant background NS–NS B-field is equivalent to a (p+1)-dimensional non-commutative field theory whose non-commutative (constant) parameter θij is given by the background B-field Bij.

It is a well-known fact that the theory with a constant second-rank anti-symmetric tensor cannot have explicit Lorentz invariance in p+1 dimensions for p⩾2. In string theory, NS–NS B-field is a dynamical field and its constant background field can be regarded as the vacuum expectation value of the B-field. In this view point, Lorentz symmetry is spontaneously broken by the vacuum expectation value of the second rank anti-symmetric tensor field.

One can ask naturally how Lorentz symmetry is realized in the broken theory and what is the Nambu–Goldstone boson for the spontaneous Lorentz symmetry breaking. In this paper, we study the first problem. We also discuss the low-energy effective action of the corresponding Nambu–Goldstone bosons, which is obtained from only the symmetry argument. The second problem, which is model-dependent, will be studied in the forthcoming paper [4].

This paper is organized as follows. In Section 2, we summarize the nonlinear realization of general space–time symmetry. In Section 3, we study the nonlinear realization of the Lorentz symmetry and construct the effective action of Nambu–Goldstone bosons and other fields invariant under the nonlinear transformation of the Lorentz symmetry. In Section 4, we discuss the related topics.

Section snippets

Nonlinear realization of space–time symmetry

It is well-known that if a symmetry is broken spontaneously, the broken symmetry is realized nonlinearly in the effective theory. In this section we summarize the general theory of the nonlinear realization of space–time symmetry, such as conformal symmetry, supersymmetry and so on, following [5] (references therein). We will apply this formalism to the Lorentz symmetry in the next section.

Let G be a group of a space–time symmetry and H be its stability subgroup, i.e., unbroken subgroup.1

Nonlinear realization of the Lorentz symmetry

In this section, as a simplest example, we consider the nonlinear realization of the Lorentz symmetry in 2+1 dimensions. In 2+1 dimensions, the Poincaré algebra (∼iso(2,1)) is given by the Lorentz generators Mμν and the translation generators Pμ:4 Pμ,Pν=0,Jμ,Pν=−iϵμνρPρ,Jμ,Jν=−iϵμνρJρ,whereJμ12ϵμνρMνρ(μ=0,1,2). The homogeneous Lorentz subgroup generated by Jμ forms SO(2,1)∼SL(2,R) and we take a basis as J0=12σ3,J1=i2σ1,andJ2=i2σ2, where σi's are Pauli

Discussions

We have discussed the nonlinear realization of the Lorentz symmetry in 2+1 dimensions and obtained the low-energy effective action invariant under the nonlinear transformation of the Lorentz symmetry.

We comment shortly on the several examples that realize the spontaneous Lorentz symmetry breaking. The first example is gauge invariant field theory of the second rank anti-symmetric tensor field Bμν [10], [11]. In the free field theory, the vev of Bμν can be a non-zero constant, thus the vev (3.4)

Acknowledgements

The author is very grateful to K. Higashijima for the useful suggestions and discussions and careful reading of this manuscript. The author also thanks Y. Hosotani for the useful discussions.

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