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Renormalization of a Deformed Scalar Quantum Field Theory

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Renormalization of a Deformed Scalar Quantum Field Theory PHYSICAL REVIEW D 69, 085003 ͑2004͒ Renormalization of a deformed scalar quantum field theory V. B. Bezerra* Departamento de Fı´sica, Universidade Federal da Paraı´ba, Caixa Postal 5008, 58051-970 - Joa˜o Pessoa, PB, Brazil E. M. F. Curado† and M. A. Rego-Monteiro‡ Centro Brasileiro de Pesquisas Fı´sicas, Rua Xavier Sigaud 150, 22290-180 - Rio de Janeiro, RJ, Brazil ͑Received 15 September 2003; published 6 April 2004͒ In this paper we consider a q-deformed scalar quantum field theory and show that this theory is renormal- izable up to second order in the coupling constant. This result is obtained in a similar way to the nondeformed standard theory. DOI: 10.1103/PhysRevD.69.085003 PACS number͑s͒: 03.70.ϩk, 11.10.Ef, 11.10.Gh I. INTRODUCTION and relativistic ͓8͔ square-well potentials in one dimension and the harmonic oscillator on a circle ͓9͔. The introduction of the hypothesis that radiation is emit- Towards the construction of a second quantized phenom- ted in certain quanta in 1900 by Planck is considered as the enological approach to models with composite particles we birth of quantum physics. After that it took more than twenty constructed in ͓10,11͔ a quantum field theory ͑QFT͒ which years to understand, starting from the application of the con- creates particles described by a q-deformed Heisenberg alge- cepts of quantum physics to radiation, that particles are de- bra. These particles are created and/or annihilated by opera- scribed by fields which can be seen as a superposition of tors satisfying a deformed Heisenberg algebra with a nonad- quantum harmonic oscillators. One could say that, in the ditive energy spectrum. As this algebra approximates a second quantization formalism, Planck’s hypothesis is real- particular type of Heisenberg algebra describing a composite ized by the use of creation and annihilation operators which system ͓2͔, we interpret that QFT as a phenomenological satisfy Heisenberg algebra. theory describing the interaction of composite particles. In In addition to the idea that the energy of a point particle is our first paper ͓10͔, we constructed the Dyson-Wick contrac- emitted in certain quanta, it also happens that its energy is tion of two fields, for the free theory and computed pertur- additive since the energy of n point particles is given by n batively the scattering 1ϩ2!1Јϩ2Ј to first order in the times the energy of one point particle. However, we know, at coupling constant. In the second ͓11͔, we discussed the gen- least from the middle of the last century ͓1͔, that in the case eralization of Wick’s expansion and computed perturbatively of composite particles Planck’s postulate remains valid but the same process considered in ͓10͔, but now up to the sec- the additivity rule for energies is lost. ond order in the coupling constant. Together with the nonadditivity of energy, it was realized The purpose of this paper is to start the analysis of the that the creation and annihilation operators of composite par- renormalizability of the deformed QFT we discussed in Refs. ticles satisfy different forms of Heisenberg algebra. In 1992 ͓10͔ and ͓11͔. This paper is organized as follows: In Sec. II, Bonatsos ͓2͔ showed that correlated fermion pairs, in simple we summarize the linear case of the GHA. In Sec. III, we many body systems, satisfy a deformed Heisenberg algebra present some remarks concerning a q-deformed quantum which could be approximated by a deformation of Heisen- field theory. In Sec. IV, we show that this deformed theory is berg algebra known as q oscillators ͓3͔. This suggests that it renormalizable up to second order in the coupling constant. could be interesting to use q-oscillator algebra as a phenom- Finally, in Sec. V, we end with our final comments. enological approach to treat models with composite particles. Interesting generalizations of Heisenberg algebras have II. LINEAR „q-OSCILLATORS… GENERALIZED been considered in past years ͓4͔. Recently a class of alge- HEISENBERG ALGEBRA bras was constructed called generalized Heisenberg algebras ͑GHA͒, having as generators the creation, annihilation opera- Recently, we have introduced an algebra, called general- ͓ ͔ ized Heisenberg algebra, which is generated by the operators tors and the Hamiltonian of a quantum system 5–9 . The † ͓ ͔ GHA describes a class of quantum systems characterized by J0 , A and A , obeying the commutation relations 5,6 ⑀ having successive energy eigenvalues obeying nϩ1 J A†ϭA† f ͑J ͒, ͑1͒ ϭ ⑀ 0 0 f ( n), with the function f (x) being a characteristic func- tion of the algebra. As examples of systems which are de- ϭ ͒ ͑ ͒ AJ0 f ͑J0 A, 2 scribed by a GHA with an appropriated function f (x) we can ͓ ͔ † ϭ Ϫ ͒ ͑ ͒ mention the above referred q oscillators, non-relativistic 7 ͓A ,A͔ J0 f ͑J0 . 3 Note that these operators satisfies, by hypothesis, the rela- † †ϭ †ϭ *Email address: valdir@fisica.ufpb.br tions (A ) A, J0 J0, and f (x) is, in general, an analytical †Email address: [email protected] function of x. It was shown in Refs. ͓7–9͔ that A†, A are the ‡ Email address: [email protected] creation, annihilation operators and J0 is the Hamiltonian of 0556-2821/2004/69͑8͒/085003͑7͒/$22.5069 085003-1 ©2004 The American Physical Society BEZERRA, CURADO, AND REGO-MONTEIRO PHYSICAL REVIEW D 69, 085003 ͑2004͒ † ͑ ͒ ͑ ͒ the physical system under consideration. When the function f tors J0 , A and A defined above satisfy Eqs. 1 – 3 which is linear, we will write it, for practical reasons as f (x) can be verified if Eqs. ͑4͒, ͑5͒ are used. 2 ϭq xϩs where q and sϾ0 are real numbers. In this case, An interpretation of this deformed Heisenberg algebra as ͑ ͒ ͑ ͒ ! Eqs. 1 – 3 can be rewritten, after the rescaling J0 sJ0 , an approximative tool to describe composite particles was A!A/ͱs and A†!A†/ͱs,as developed in the past few years ͓10,11,14͔ and is based on † ϭ † ͑ ͒ the commutation relations of composite bosonlike particles ͓J0 ,A ͔q2 A , 4 † † constituted by two fermions. If a1 and a2 are the creation operators of fermions labeled by the numbers 1 and 2, the 1 creation operator of a composite ͑by these two fermions͒ ͓J ,A͔ Ϫ2ϭϪ A, ͑5͒ 0 q 2 †ϭ † † q bosonlike particle could be written as b a2a1 ; the annihi- lation operator of this bosonlike particle is its adjoint. Defin- † ϭ Ϫ 2͒ Ϫ ͑ ͒ ͓A ,A͔ ͑1 q J0 1, 6 ing the number operator of the bosonlike particles as N ϭ(a†a ϩa†a )/2, the commutation relations of N, b and b† ϵ Ϫ 1 1 2 2 where ͓a,b͔r ab rba is the r-deformed commutation of ! obey a deformed Heisenberg algebra which can be obtained the operators a and b. Thus, we have that when q 1we as a particular case of the q-oscillator algebra. As a physical р Ͻϱ Þ reobtain the Heisenberg algebra and when 0 q , q 1, example of this, Bonatsos showed that the algebra of fermion this algebra can be mapped into the q-oscillator algebra. pairs with zero angular momentum can be approximated by Nonlinear functions can also be considered and an analysis the q-oscillator algebra ͓2͔. Also, their pairing Hamiltonian ͓ ͔ for quadratic functions can be seen in 5–7 . has a nonadditivity property, in the sense that the energy Using a noncommutative differential calculus, defined on difference of any two successive levels are not equal. This is a discrete lattice ͓12,13͔, it is possible to relate the operators † a property of deformed Heisenberg algebras unlike the har- J0 , A and A of this algebra with the physical operators monic oscillator algebra. In their shell model of nuclear col- position and momentum. To illustrate this noncommutative lective motion ͓2͔, the fermion pairs of zero angular momen- discrete differential calculus and to show the connection of tum (Jϭ0) in a single j-shell are created by the pair creation the operators of the algebra with the physical operators, let operator us define a one-dimensional lattice in a momentum space, with a spacing parameter a. The right and left derivatives of an arbitrary function g on these lattices can be defined 1 †ϭ ͚ ͑Ϫ ͒ jϩm † † ͑ ͒ B 1 f j,m f j,Ϫm , 13 ץ¯ ϭ ϩ Ϫ ץ as ( pg)(p) (1/a)͓g(p a) g(p)͔ and ( pg)(p) ͱ⍀ mϾ0 ϭ(1/a)͓g(p)Ϫg(pϪa)͔. Defining the shift momentum op- ץ¯ ϭ Ϫ ¯ ץ ϭ ϩ erators T 1 a p and T 1 a p , that changes the mo- Ϫ р р † with j m j, where f j,m are fermion creation operators mentum value by a, i.e., (Tg)(p)ϭg(pϩa) and (¯Tg)(p) and 2⍀ϭ2 jϩ1 is the degeneracy of the shell ͑here we are ϭg(pϪa), satisfying TT¯ ϭ¯TTϭ1ˆ , where 1ˆ represents the using the same notation of Ref. ͓2͔͒. The pair creation and unit in the algebra of the arbitrary functions of p,wecan annihilation operators satisfy the algebra introduce the momentum operator P ͓12,13͔ satisfying (Pg)(p)ϭpg(p). The operators P, T and ¯T satisfy the com- N ͓ †͔ϭ Ϫ F ͑ ͒ mutation relations B,B 1 ⍀ , 14 ͓T,P͔ϭPϩa, ͑7͒ ϭ͚ † ϩ † where NF mϾ0( f j,m f j,m f j,Ϫm f j,Ϫm) is the fermion ͓¯T,P͔ϭPϪa.
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