Tsallis' Quantum Q-Fields

Chinese Physics C Vol. 42, No. 5 (2018) 053102

Tsallis’ quantum q-ﬁelds

A. Plastino1,3,4 M. C. Rocca1,2,3 1 Departamento de F´ısica, Universidad Nacional de La Plata, Argentina 2 Departamento de Matemática, Universidad Nacional de La Plata, Argentina 3 Consejo Nacional de Investigaciones Cient´ıficas y Tecnológicas (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 fields associated with them in the recent literature. We refer to the q-Schrödinger, 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 fields, both in the Abelian and non-Abelian instances. We show how to define the q-quantum field theories corresponding to the above equations, introduce the pertinent actions, and obtain equations of motion via the minimum action principle. These q-fields 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-fields are simultaneously q-exponential functions of the usual linear fields’ logarithms.

Keywords: non-linear Klein-Gordon, non-linear Schrödinger and non-linear q-Dirac fields, non-linear q-Yang-Mills and non-linear q-Proca fields, classical field theory, quantum field 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 field 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 field theories (q-QFTs) have also Our new quantum field theory corresponds to non- been discussed [2, 3]. linear equations. Thus, gauge and Lorentz invariance are These CFTs cannot be directly quantified because of broken. These invariances reappear in the limit q 1. µ → non-linearity, which means the superposition principle A nice property of our new equation ∂µA = 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-fields. We will fields, it is also valid for q-Abelian Yang-Mills and q- here remedy such a formidable quantification obstacle by Proca fields. 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 fields, 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 quantification of these two coupled fields is Schrödinger, 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) defined in Ref. [5]. Also, and tions can be divided into two types: (A) basic equations for the first 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 field. chanics, mainly at the boundary between quantum and It has been shown in Refs. [7, 8] that q-fields 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 Schrödinger-like gies (GeV) for q =1.001, and 3) low energies (MeV) for equations (NLSE) as effective, single particle mean field q=1.000001. The ALICE experiment at the LHC shows descriptions of involved quantum many-body systems. A that Tsallis q-effects 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

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2 Non-linear q-Schrödinger field treatment of fields given in Ref. [1]. Accordingly, we can obtain quantum q-fields starting from the usual q = 1

We proceed now to effect a transformation that re- quantum fields. We can also express ψ in terms of ψq as quires some previous considerations. Consider two dif- (1−q) ψq −1 ferent formalisms A and B that can be connected by an ψ=e 1−q (5) appropriate mathematical transformation. Assume that The Schrödinger field 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 † † 3 = i~ψ ∂tψ ψ ψ dtd x. (6) ism B, that could be mathematically more involved than S Z µ −2m∇ ∇ ¶ 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~∂ ψ+ ψ=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 find q-exponentials, ψ(~x,t)= a(p~)e ~ (p~·~x−Et)d3p. (8) π~ 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 field ψq is example refers to the Schrödinger equation (SE) with (1−q) †(1−q) ψq −1 ψq −1 q variable mass, that has multiple applications. Here there 1−q 1−q − q = e e ψq exists a transformation connecting the SE with constant S Z ~2 mass with the SE with variable mass. Why bother with q i~∂ ψ ψ†− ψ† ψ dtd3x, (9) such a transformation? Answer: in many problems in ×µ t −2m q ∇ q ∇ q¶ solid state physics, nuclear physics, etc., the relevant constructed keeping in mind that the field ψq satisfies physics is described by the SE with variable mass. The ~2 transformation that we are advancing here reads: ~ 2 −q −1 i ∂tψq+ [ ψq+( ψq) (ψq qψq )]=0. (10) 1 2m 4 ∇ − ψ (~x,t)=[1+(1 q)lnψ(~x,t)] 1−q , (1) q − + Note that the q-exponential wave (6) is, by construction, where ψ is the usual quantum Schrödinger field 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 Schrödinger equation. The same is true cut-off. 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-off has no relevance since ψ the usual one and that Eq. (10) is the q-generalization is an operator and the information is contained in the of the ordinary Schrödinger equation. pertinent operators of creation and annihilation. At this Additionally, since in Eq. (1) the field ψ is a quan- level we have: tum field, this implies that ψq is of such a nature too.

1 1 Of course, for q 1, ψ becomes ψ. Physically, if the 1−q ln{[I+(1−q)lnψ]} q ψ = [I+(1 q)lnψ] =e 1−q → q − + energy goes down, the q-field transforms itself into the ∞ n φ (q 1) 2 usual one (remember our assertions above on the connec- = an =I+φ+ − φ +... , (2) n! 2 tion between q-fields and the energy scale based on the Xn=0 work at ALICE at the LHC [7, 8]). Given that we speak where ψ=I+φ. There are no cuts or branch points then. here of a non-linear QFT, direct field quantification 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 ψ 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) ψ(~x,t)=e ~ . (3) as we did here, an indirect route is necessary to quantify Replacing this into Eq. (1), we find a classical field.

1 i 1−q ψq(~x,t)=[1+(1 q) (p~ ~x Et)]+ . (4) 3 Non-linear q-Klein-Gordon (KG) field − ~ · − This is just the q-wave that Nobre et al. [1] used to In the same vein as above, we define a quantum q- obtain the q-Schrödinger, q-Klein-Gordon, and q-Dirac Klein Gordon (KG) field φq(xµ) in terms of the ordinary equations. Thus, the q-wave is a particular case of the KG field φ(xµ) as quantum field defined by Eq. (1). This allows for imme- 1 diate generalization to the quantum realm of the classical φ (x )=[1+(1 q)lnφ(x )] 1−q . (11) q µ − µ +

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In the classical instance, if we have that, of course, are the Dirac and Klein-Gordon equa-

i(~k·~x−ωt) tions, respectively. We define now a very high energy φ(xµ)=e , (12) field ψqa as 1 we re-obtain the q-wave used by Nobre et al. in Ref. [1]: 1−q ψqa =[1+(1 q)lnψa]+ , (25) 1 − 1−q φ (x )=[1+(1 q)i(~k ~x ωt)] . (13) not Lorentz invariant. ψqa is not a component of the q µ − · − + Dirac-spinor. Let us now cast ψa in terms of ψqa: φ can be given in terms of φq as (1−q) ψqa −1 −q (1 ) 1−q φq −1 ψa =e . (26) φ=e 1−q . (14) The ψqa-associated action is From Ref. (11) we see that φq is not Lorentz invariant †(1−q) (1−q) ψ −1 ψ −1 (LI). We saw above that it manifests itself at very high qa †(−q) qb 1−q 0 µ 1−q q = ie (γ γ )abψ ∂µψqbe energy. If the energy becomes smaller, and this happens S Z qb Xab for q 1, φq becomes φ and LI is restored. The usual (1−q) †(1−q) ψ −1 → ψqa −1 qb KG action is 0 4 me 1−q γ e 1−q d x . (27) − ab µ µ † 2 † 4 = ∂µφ(xµ)∂ φ (xµ) m φ(xµ)φ (xµ) d xµ, (15) Given the lack of Lorentz invariance, Einstein’s conven- S Z − £ ¤ tion on repeated indexes cannot be used. This action from which one deduces becomes that of Eq. (22) for q 1. From Eq. (27) one → (¤+m2)φ=0, (16) deduces the equations of motion for ψq as (1−q) (1−q) ψ −1 ψ −1 µ q qb qa whose solution is − 1−q 1−q iγabψqb ∂µψqbe me =0 (28) ~ † ~ − 1 a(k) i(~k·~x−ωt) a (k) −i(~k·~x−ωt) 3 ¤ µ −q −1 2 q φ(xµ)= 3 e + e d k, ψqa+∂µψqa∂ ψqa(ψqa qψqa )+m ψqa =0, (29) (2π) 2 Z √2ω √2ω − that become Eq. (23) – Eq. (24) when q 1. The energy (17) → this being the field φ in Eq. (11). For φq one has considerations in this limit made above in the KG case also hold here. (1−q) †(1−q) φq −1 φq −1 1−q 1−q −q †−q † 2 4 q = e e φ φ ∂µφq∂µφ m d xµ, S Z q q q− ¡ ¢ 5 Advancing a non-linear Abelian Yang- (18) Mills q-field leading to an equation of motion whose solution is φq, that is It is well known that the action for an Abelian Yang- ¤φ +∂ φ ∂µφ (φ−q qφ−1)+m2φq =0. (19) Mills field reads q µ q q q − q q 1 µν 4 For q 1, Eq. (18) becomes Eq. (15) while Eq. (19) goes = µν d x, (30) to Eq.→(16). S −4 Z F F where =∂ A ∂ A , (31) 4 Non-linear q-Dirac field Fµν µ ν − ν µ and the associated equation of motion is Dirac’s action is known to be: µν ∂µ =0, (32) = iψ∂/ψ mψψd4x, (20) F S Z − which can be recast as two equations or µ ¤Aµ =0 ∂µA =0. (33) † 0 µ † 0 4 = iψ γ γ ∂µψ mψ γ ψd x. (21) S Z − Our present q-extension begins by defining In terms of Dirac’s spinor ψ this action is 1 A =[1+(1 q)lnA ] 1−q , (34) qµ − µ + † 0 µ † 0 4 = iψ (γ γ )ab∂µψb mψ γ ψbd x. (22) breaking Lorentz invariance (LI) once again. Conversely, S Z a − a ab we can write (1−q) We deduce now that the spinor’s components obey the Aqµ −1 1−q equations of motion Aµ =e , (35) µ leading to iγab∂µψb mψa =0, (23) − (1−q) Aqµ −1 ¤ 2 µ 1−q −q µ ( +m )ψa =0, (24) ∂ Aµ =e Aqµ ∂ Aqµ =0, (36)

053102-3 Chinese Physics C Vol. 42, No. 5 (2018) 053102 and then the action becomes µ ∂ Aqµ =0, (37) 1 µνC 4 = µνC d x, (48) S −4g2 Z F F so that the field Aqµ fulfills the Lorentz gauge, a surpris- ing result given the above LI-breaking. Our associated leading to the equation of motion q-action Aqµ is ρσD ρσC C A ∂ρ g fADAρ =0. (49) (1−q) F − F 1 Aqη −1 µρ νη 1−q −q q = g g e A ∂ρAqη We now define our q-extension S −4 Z · qη µ,νX,ρ,η 1 C C 1−q (1−q) Aqρ −1 Aqµ =[1+(1 q)lnAµ ]+ , (50) 1−q −q − e A ∂ηAqρ − qρ ¸ again breaking both Lorentz and gauge invariance for (1−q) (1−q) Aqν −1 Aqµ −1 q>1. From Eq. (50) we obtain q q e 1−q A− ∂ A e 1−q A− ∂ A d4x, qν µ qν qµ ν qµ C ⊗· − ¸ Aqµ−1 C A =e 1−q , (51) (38) µ and the action associated with the field (50) is leading to the equation of motion

ν −q −1 1 µµ νν C C 4 ¤Aqµ+∂ν Aqµ∂ Aqµ(A qA )=0, (39) = g g d x, (52) qµ − qµ S −4g2 Z Fqµν Fqµν µ,νX,C obeyed by Aqµ. It is clear that for q 1 our new theory becomes the customary Abelian Yang-Mills→ theory. where C C Aqν −1 Aqµ−1 C qC C qC C = A− e 1−q ∂ A A− e 1−q ∂ A 6 Introducing our non-linear non- Fqµν qν µ qν − qµ ν qµ A B Aqµ−1 Aqν −1 Abelian Yang-Mills q-field C 1−q 1−q +gfAB e e , (53) XA,B The Yang-Mills theory is a gauge theory, constructed from a Lie algebra, that attempts to describe the behav- leading to the equation of motion A ior of elementary particles via non-Abelian Lie groups. Aqρ−1 ρρ D ρρ C C 1−q This lies at the core of i) the unification of the weak and g ∂ρ qρσ g g qρσfADe =0. (54) F − F electromagnetic forces, as well as ii) quantum chromody- Xρ ρACX namics. It constitutes the foundation of our understand- C The field Aqµ satisfies this equation, of course. Whenever ing of the standard model. The corresponding action the energy becomes low enough, q 1, and one recovers is LI and gauge invariance. → 1 = tr(F µν F )d4x, (40) 2g2 µν S − Z 7 Our non-linear quantum Proca q-field where F µν µν C = C T . (41) The Proca action gives a detailed account of a mas- F sive spin-1 field of mass m in a Minkowskian space-time. Here the matrices T C correspond to a non-Abelian, semi- The associated equation is a relativistic-wave equation, simple Lie group. One has called the Proca equation. The action is C C [TA,TB ]=fAB T , (42) 1 †µν 2 † µ 4 = µν 2m AµA d x, (55) δAB S −2 Z F F − tr(TATB )= , (43) 2 where where F µν is =∂ A ∂ A . (56) Fµν µ ν − ν µ F µν =∂ A ∂ A ig[A ,A ], (44) µ ν − ν µ − µ ν the equations of motion being 2 µ with (¤+m )Aµ =0 ∂µA =0. (57) A C µ =Aµ TC (45) At this stage we define our q-action and 1 1−q C =∂ AC ∂ AC +gf C AAAB . (46) Aqµ =[1+(1 q)lnAµ] . (58) Fµν µ ν − ν µ AB µ ν − Because of the relation breaking LI. Inversion of Eq. (58) gives (1−q) F µν F 1 µνC Aqµ −1 tr( µν )= µνC , (47) A =e 1−q (59) 2F F µ

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From the second relation in Eq. (57) and from Eq. (59) Refs. [1–6]. We have also added equations corresponding we find to q-Yang-Mills fields, both Abelian and non-Abelian. µ ∂ Aqµ =0, (60) We have obtained the q-quantum field theories cor- whose associated action is responding to all the above equations, and showed that in the limit q 1 they become the customary ones. 1 µµ νν †µν → = g g q qµν These results agree with our previous results [7, 8] S −2 Z F F Xµ,ν concerning the energies involved. One needs energies of †(1−q) (1−q) Aqµ −1 Aqµ −1 up to 1 TeV in order to clearly distinguish between q- 2m2 e 1−q e 1−q d4x, (61) − theories and q=1, ordinary ones. Xµ All our new quantum q-fields are q-exponential func- with tions of the logarithms of the conventional q = 1 fields. (1−q) (1−q) Aqν −1 Aqµ −1 q q We have seen that these cannot be directly quantified =A− e 1−q ∂ A A− e 1−q ∂ A . (62) Fqµν qν µ qν − qµ ν qµ because of non-linearity, which makes the superposition From both this and Eq. (60) one finds the equation of principle non-applicable, and then it is impossible to in- motion troduce creation/annihilation operators for the q-fields. To remedy such a formidable quantification obstacle we ¤A +(A−q qA−1) gνν (∂ A )2+m2Aq =0, (63) qµ qµ − qµ ν qµ qµ have here devised an indirect approach that has been Xν shown to work correctly. satisfied by A . LI is recovered in the limit q 1. qµ → An interesting fact is that a Tsallis’ q-exponential wave is a solution of the equations of motion (10), (19), 8 Conclusions (28), (29), (39), (54), and (63), although these all look quite different! We have obtained some new quantum results that may be regarded as interesting. We thank Prof. A. R. Plastino for helpful dis- More specifically, we have generalized to the quan- cussions. We are indebted to CONICET (Argentine tum realm the classical Tsallis’ q-Schrödinger, q-Klein- Agency) for economic support. Gordon, q-Dirac, and q-Proca equations obtained in

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