Chinese Physics C Vol. 42, No. 5 (2018) 053102 Tsallis' quantum q-¯elds A. Plastino1;3;4 M. C. Rocca1;2;3 1 Departamento de F¶³sica, Universidad Nacional de La Plata, Argentina 2 Departamento de Matem¶atica, Universidad Nacional de La Plata, Argentina 3 Consejo Nacional de Investigaciones Cient¶³¯cas y Tecnol¶ogicas (IFLP-CCT-CONICET) -C. C. 727, 1900 La Plata - Argentina 4 SThAR - EPFL, Lausanne, Switzerland Abstract: We generalize several well known quantum equations to a Tsallis' q-scenario, and provide a quantum version of some classical ¯elds associated with them in the recent literature. We refer to the q-SchrÄodinger, q-Klein- Gordon, q-Dirac, and q-Proca equations advanced in, respectively, Phys. Rev. Lett. 106, 140601 (2011), EPL 118, 61004 (2017) and references therein. We also introduce here equations corresponding to q-Yang-Mills ¯elds, both in the Abelian and non-Abelian instances. We show how to de¯ne the q-quantum ¯eld theories corresponding to the above equations, introduce the pertinent actions, and obtain equations of motion via the minimum action principle. These q-¯elds are meaningful at very high energies (TeV scale) for q =1:15, high energies (GeV scale) for q =1:001, and low energies (MeV scale) for q=1:000001 [Nucl. Phys. A 955 (2016) 16 and references therein]. (See the ALICE experiment at the LHC). Surprisingly enough, these q-¯elds are simultaneously q-exponential functions of the usual linear ¯elds' logarithms. Keywords: non-linear Klein-Gordon, non-linear SchrÄodinger and non-linear q-Dirac ¯elds, non-linear q-Yang-Mills and non-linear q-Proca ¯elds, classical ¯eld theory, quantum ¯eld theory PACS: 11.10.Ef, 11.10.Lm, 02.30.Jm DOI: 10.1088/1674-1137/42/5/053102 1 Introduction ergies. We will see that all q-QFTs employed here transform Classical ¯eld theories (CFT) associated with Tsallis' into the well known associated QFTs for q 1, entailing ! q-scenarios have received much attention recently [1{6]. going down from extremely high energies to lower ones. Associated q-quantum ¯eld theories (q-QFTs) have also Our new quantum ¯eld theory corresponds to non- been discussed [2, 3]. linear equations. Thus, gauge and Lorentz invariance are These CFTs cannot be directly quanti¯ed because of broken. These invariances reappear in the limit q 1. ¹ ! non-linearity, which means the superposition principle A nice property of our new equation @¹A = 0, is that is not applicable, and it is then impossible to introduce as well as being valid for Abelian Yang-Mills and Proca creation/annihilation operators for the q-¯elds. We will ¯elds, it is also valid for q-Abelian Yang-Mills and q- here remedy such a formidable quanti¯cation obstacle by Proca ¯elds. recourse to an indirect approach. M.A. Rego-Monteiro et al. [6] have tackled in recent In this paper we both extend to the quantum realm years the possible need for two coupled ¯elds, instead of and generalize several aspects of the above mentioned only one, to properly handle classical non-linear equa- works. We construct the CFTs corresponding to the q- tions. The quanti¯cation of these two coupled ¯elds is SchrÄodinger, q-Klein-Gordon, and q-Dirac equations in- discussed in Refs. [2] and [3]. troduced in Refs. [1{4]. We do the same for the q-Proca Motivations for non-linear quantum evolution equa- and q-Yang-Mills (Abelian) de¯ned in Ref. [5]. Also, and tions can be divided into two types: (A) basic equations for the ¯rst time ever, we deal with the equation and q- governing phenomena at the frontiers of quantum me- QFT corresponding to a non-Abelian q-Yang-Mills ¯eld. chanics, mainly at the boundary between quantum and It has been shown in Refs. [7, 8] that q-¯elds emerge at gravitational physics (see Refs. [10, 11] and references 1) very high energies (TeV) for q = 1:15, 2) high ener- therein); and (B) regarding non-linear SchrÄodinger-like gies (GeV) for q =1:001, and 3) low energies (MeV) for equations (NLSE) as e®ective, single particle mean ¯eld q=1:000001. The ALICE experiment at the LHC shows descriptions of involved quantum many-body systems. A that Tsallis q-e®ects manifest themselves [9] at TeV en- paradigmatic illustration is that of Ref. [12]. Received 16 December 2017, Revised 21 February 2018, Published online 17 April 2018 ©2018 Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd 053102-1 Chinese Physics C Vol. 42, No. 5 (2018) 053102 2 Non-linear q-SchrÄodinger ¯eld treatment of ¯elds given in Ref. [1]. Accordingly, we can obtain quantum q-¯elds starting from the usual q = 1 We proceed now to e®ect a transformation that re- quantum ¯elds. We can also express à in terms of Ãq as quires some previous considerations. Consider two dif- (1¡q) Ãq ¡1 ferent formalisms A and B that can be connected by an Ã=e 1¡q (5) appropriate mathematical transformation. Assume that The SchrÄodinger ¯eld action is well known as we know how to solve the relevant equations for A. A S legitimate question is why to bother at all with formal- ~2 y y 3 = i~à @tà à à dtd x: (6) ism B, that could be mathematically more involved than S Z µ ¡2mr r ¶ A. The answer is as follows. Even though A and B are mathematically connected, it is possible that, in some From it one deduces the equation of motion scenarios, the variables in B provide a more appropriate ~2 i~@ Ã+ Ã=0; (7) description of some natural phenomenon. There is some t 2m4 experimental evidence that such is the case with Tsallis- whose solution is inspired non-linear wave equations (the ALICE experi- 1 i ment at CERN). Empirically, they ¯nd q-exponentials, Ã(~x;t)= a(p~)e ~ (p~¢~x¡Et)d3p: (8) ¼~ 3 that are solutions to the q-equations of motion, suggest- (2 ) 2 Z ing that Nature uses the non-standard scenario. Another The action corresponding to the ¯eld Ãq is example refers to the SchrÄodinger equation (SE) with (1¡q) y(1¡q) Ãq ¡1 Ãq ¡1 q variable mass, that has multiple applications. Here there 1¡q 1¡q ¡ q = e e Ãq exists a transformation connecting the SE with constant S Z ~2 mass with the SE with variable mass. Why bother with q i~@ à Ãy¡ Ãy à dtd3x; (9) such a transformation? Answer: in many problems in £µ t ¡2m q r q r q¶ solid state physics, nuclear physics, etc., the relevant constructed keeping in mind that the ¯eld Ãq satis¯es physics is described by the SE with variable mass. The ~2 transformation that we are advancing here reads: ~ 2 ¡q ¡1 i @tÃq+ [ Ãq+( Ãq) (Ãq qÃq )]=0: (10) 1 2m 4 r ¡ à (~x;t)=[1+(1 q)lnÃ(~x;t)] 1¡q ; (1) q ¡ + Note that the q-exponential wave (6) is, by construction, where à is the usual quantum SchrÄodinger ¯eld opera- a solution to Eq. (10). For q 1 this last equation be- ! tor, and the subscript + indicates the so-called Tsallis comes the usual SchrÄodinger equation. The same is true cut-o®. for the action given by Eq. (9). One is then in a position At a quantum level, which is the case that we are to assert that such an action is the q-generalization of interested in here, the cut-o® has no relevance since à the usual one and that Eq. (10) is the q-generalization is an operator and the information is contained in the of the ordinary SchrÄodinger equation. pertinent operators of creation and annihilation. At this Additionally, since in Eq. (1) the ¯eld à is a quan- level we have: tum ¯eld, this implies that Ãq is of such a nature too. 1 1 Of course, for q 1, à becomes Ã. Physically, if the 1¡q lnf[I+(1¡q)lnÃ]g q à = [I+(1 q)lnÃ] =e 1¡q ! q ¡ + energy goes down, the q-¯eld transforms itself into the 1 n Á (q 1) 2 usual one (remember our assertions above on the connec- = an =I+Á+ ¡ Á +::: ; (2) n! 2 tion between q-¯elds and the energy scale based on the Xn=0 work at ALICE at the LHC [7, 8]). Given that we speak where Ã=I+Á. There are no cuts or branch points then. here of a non-linear QFT, direct ¯eld quanti¯cation by No information is lost if one considers the whole series. appeal to creation-destruction operators is not feasible, Consider now the classical instance in which à is just since the superposition principle is no longer valid. The a plane wave reasoning applies to the propagator notion as well. Thus, i (p~¢~x¡Et) Ã(~x;t)=e ~ : (3) as we did here, an indirect route is necessary to quantify Replacing this into Eq. (1), we ¯nd a classical ¯eld. 1 i 1¡q Ãq(~x;t)=[1+(1 q) (p~ ~x Et)]+ : (4) 3 Non-linear q-Klein-Gordon (KG) ¯eld ¡ ~ ¢ ¡ This is just the q-wave that Nobre et al. [1] used to In the same vein as above, we de¯ne a quantum q- obtain the q-SchrÄodinger, q-Klein-Gordon, and q-Dirac Klein Gordon (KG) ¯eld Áq(x¹) in terms of the ordinary equations. Thus, the q-wave is a particular case of the KG ¯eld Á(x¹) as quantum ¯eld de¯ned by Eq.