Quantum q-Field Theory: q-Schrödinger and q-Klein-Gordon Fields
aa r X i v : . [ h e p - t h ] A ug Quantum q-Field Theory:q-Schr¨odinger and q-Klein-GordonFields
A. Plastino and M. C. Rocca La Plata National University and Argentina’s National Research Council(IFLP-CCT-CONICET)-C. C. 727, 1900 La Plata - Argentina
September 1, 2018
Abstract
We show how to deal with the generalized q-Schr¨odinger and q-Klein-Gordon fields in a variety of scenarios. These q-fields are mean-ingful at very high energies (TeVs) for for q = , high ones (GeVs)for q = , and low energies (MeVs)for q = [Nucl. Phys.A (2016) 19; Nucl. Phys. A (2016) 16]. (See the Alice ex-periment of LHC). We develop here the quantum field theory (QFT)for the q-Schr¨odinger and q-Klein-Gordon fields, showing that both re-duce to the customary Schr¨odinger and Klein-Gordon QFTs for q closeto unity. Further, we analyze the q-Klein-Gordon field for q ≥ .In this case for − = n (n integer ≥ ) and analytically computethe self-energy and the propagator up to second order. Keywords:
Non-linear Klein-Gordon equation; Non-linear Schr¨odingerequation; Classical Field Theory, Quantum Field Theory.
PACS: Introduction
Classical fields theories (CFT) associated to Tsallis’ q-scenarios have beenintensely studied recently [10, 9, 24]. Associated quantum (QFT) treatmentshave also been discussed [24]. In this paper we show how to treat the q-Schr¨odinger and q-Klein-Gordon fields in a variety of cases. It has been shownin [19, 20] that q-fields emerge at1) very high energies (TeV) for q = ,2) High (GeV) for q = , and 3) low (MeV) for q = . LHC-Alice experiments show that Tsallis q-effects manifest themselves [21] at TeVenergies.In this effort we develop QFTs associated to q-Schr¨odinger and q-Klein-Gordon fields. Moreover, we study the q-KG field in the case − = n, n integer ≥ . Here we evaluate the selfenergy and propagator up to ssecondorder, thus generalizing results of [24]. In this respect, note also recent workon Proca-de Broglies’ classical field theory [22].Motivations for nonlinear quantum evolution equations can be divided upinto two types, namely, (A) as basic equations governing phenomena at thefrontiers of quantum mechanics, mainly at the boundary between quantumand gravitational physics (see, [[1, 2] and references therein). The other possi-bility is (B) regard nonlinear-Schr¨odinger-like equations (NLSE) as effective,single particle mean field descriptions of involved quantum many-body sys-tems. A paradigmatic illustration is that of [3]. In earlier applications ofnonlinear Schr¨odinger equations, one encounters situations involving a cubicnonlinearity in the wave function.Referring to (A), our present NLSE can be used for a description of darkmatter components, since the associated variational principle (the one thatleads to the NLSE) is seen to describe particles that can not interact withthe electromagnetic field [4]. Withe reference to (B), we remark that theNLSE displays strong similarity with the Schr¨odinger equation linked to aparticle endowed with a time-position dependent effective mass [5, 6, 7, 8] ,involving particles moving in nonlocal potentials, reminiscent of the energydensity functional quantum many-body problem’s approach [9].During the last years, the search for insight into a number of complexphenomena produced interesting proposals involving localized solutions at-tached to non linear Klein-Gordon and Schr¨odinger equations, i.e., non lineargeneralizations of these equations [4, 10]. Following [4], we extend these gen-eralizations here by developing quantum field theories (QFT) associated tothe q-Schr¨odinger and q-Klein-Gordon equations [10].2ere, we develop first the classical field theory (CFT) associated to thatq-Schr¨odinger equation deduced in [11] from the hypergeometric differentialequation. We define the corresponding physical fields via an analogy withtreatments in string theory [13] for defining physical states of the bosonicstring. Our ensuing theory reduces to the conventional Schr¨odinger fieldtheory for q → .Secondly, we develop the QFT for that very q-Schr¨odinger equation (seealso [12]). This equation is similar but not identical to that advanced in [10].Its treatment is however much simpler than that employed in [4].In the third place, we develop the QFT for the q-K-G Field in severalscenarios, generalizing results of [24] and showing that the ensuing q-K-Gfield reduces to the customary K-G field for q → . We develop here the CFT for that particular q-Schr¨odinger Equation ad-vanced in [12] from the Hypergeometric Differential Equation. This NLSE isdifferent from the pioneer one proposed in [10], but exhibits better qualitativefeatures. One has i ~ ∂∂t ψ ( ~ x, t ) q = Hψ ( ~ x, t ) . (2.1)In the free particle instance one writes H = − ~ △ , (2.2)whose solution reads ψ ( ~ x, t ) = [ + ( − q ) i ~ ( ~ p · ~ x − Et )] − q . (2.3)Introduce now action S = ( − ) V ∞ Z − ∞ Z V (cid:18) i ~ ψ † q ∂ t φ † − i ~ ψ q ∂ t φ − ~ ∇ ψ ∇ φ − ∇ ψ † ∇ φ † (cid:19) dt d x, (2.4)with V the Euclidian volume. Our action can be rewritten in the fashion S = ∞ Z − ∞ Z V L ( ψ, ψ † , ∂ t φ, ∂ t φ † , ∇ ψ, ∇ ψ † , ∇ φ, ∇ φ † ) dt d x. (2.5)One obtains from (2.5) the field’s motion equations i ~ ∂∂t ψ q ( ~ k, t ) + ~ △ ψ ( ~ x, t ) = (2.6) i ~ qψ ( ~ x, t ) q − ∂∂t φ ( ~ k, t ) − ~ △ φ ( ~ x, t ) = (2.7)whose solution is (2.3). Instead, that for (2.7) reads φ ( ~ x, t ) = [ + ( − q ) i ~ ( ~ p · ~ x − Et )] − − . (2.8)If q → , φ becomes ψ † , the adjoint of ψ . Now, the concomitant canonicallyconjugated momenta are Π ψ = ∂ L ∂ ( ∂ t ψ ) = ; Π ψ † = ∂ L ∂ ( ∂ t ψ † ) = , Π φ = ∂ L ∂ ( ∂ t φ ) = − i ~ ψ q ( − ) V ; Π φ † = ∂ L ∂ ( ∂ t φ † ) = i ~ ψ † q ( − ) V , (2.9)and the associated Hamiltonian is H = Π φ ∂ t φ + Π φ † ∂ t φ † − L , (2.10)that we cast in terms of ψ - φ as H = ~ ( − ) mV (cid:0) ∇ ψ ∇ φ + ∇ ψ † ∇ φ † (cid:1) . (2.11)The field energy is E = Z V H d x. (2.12)4f we replace the solutions (2.3) and (2.8) into (2.12), one has E = Z V ~ ( − ) mV ( − ) p ~ d x, (2.13)or E = p
2m , (2.14)that exactly correspond to the wave energy (2.3), as one should expected.The field-momentum density reads ~ P = − ∂ L ∂ ( ∂ t ψ ) ∇ ψ − ∂ L ∂ ( ∂ t φ ) ∇ φ − ∂ L ∂ ( ∂ t ψ † ) ∇ ψ † − ∂ L ∂ ( ∂ t φ † ) ∇ φ † , (2.15)or ~ P = i ~ ( − ) V (cid:0) ψ q ∇ φ − ψ † q ∇ φ † (cid:1) , (2.16)the field-momentum becoming ~ P = Z V ~ P d x. (2.17)Employing (2.3) and (2.8), one finds for the momentum ~ P = i ~ ( − ) V Z V − ~ ~ pd x, (2.18)or ~ P = ~ p. (2.19)The probability density is now ρ = [ ψ q φ + ψ † q φ † ] , (2.20)verifying ∂∂t ρ + ∇ · ~ j = K, (2.21)where ~ j = ~ ( q + ) [ φ ∇ ψ − ψ ∇ φ + ψ † ∇ φ † − φ † ∇ ψ † ] , (2.22)5s the probability-current. K reads K = ~ ( q − ) [ ψ † △ φ † + φ † △ ψ † − φ △ ψ − ψ △ φ ] , (2.23)that vanishes at q = . However, the physical fields are those for which K = . For example, one lists as physical the solutions (2.3) and (2.8), sincefor them probability is indeed conserved. We start with the action S = − Z (cid:18) i ~ ψ q ∂ t φ − i ~ ψ † q ∂ t φ † + ~ ∇ ψ ∇ φ + ~ ∇ ψ † ∇ φ † (cid:19) dt d x. (2.24)We develop first a theory for 1) q close to unity and 2) weak fields ψ . Inthese condiitions one appeals to the approximation ψ q ≃ ψ + ( q − ) ψ ln ψ, (2.25)and since ψ is a weak field ψ ≃ I + ( q − ) η. (2.26)Consequently, the action (2.24) becomes S = −( q − ) Z (cid:18) i ~ η∂ t φ − i ~ η † ∂ t φ † + ~ ∇ η ∇ φ + ~ ∇ η † ∇ φ † (cid:19) dt d x, (2.27)where we used Z η ( ~ x, t ) dtd x = Z φ ( ~ x, t ) dtd x = (2.28)since the fields are η ( ~ x, t ) = ( ~ ) Z a ( ~ p ) e i ~ ( ~ p · ~ x − Et ) d p, (2.29)6see [23]) η † ( ~ x, t ) = ( ~ ) Z a † ( ~ p ) e − i ~ ( ~ p · ~ x − Et ) d p (2.30) φ ( ~ x, t ) = ( ~ ) Z b ( ~ p ) e i ~ ( ~ p · ~ x − Et ) d p, (2.31)and φ † ( ~ x, t ) = ( ~ ) Z b † ( ~ p ) e − i ~ ( ~ p · ~ x − Et ) d p. (2.32) Surprisingly enough, the q-Schr¨odinger field (qSF) reduce to the usual SF oflow energies!
Creation-destruction operators verify [ a ( ~ p ) , a † ( ~ p ′ )] = [ b ( ~ p ) , b † ( ~ p ′ )] = δ ( ~ p − ~ p ′ ) . (2.33)The propagator for the field η is [23] ∆ η ( ~ x, t ) = (cid:16) m2πi ~ (cid:17) t − + e im ~ x22 ~ t , (2.34)that, in terms of energy and momentum reads ^ ∆ η ( ~ p, E ) = i ~ E − ~ p + i0 . (2.35)These two representations are related via ∆ η ( ~ x, t ) = ( ~ ) Z ^ ∆ ( ~ p, E ) e i ~ ( ~ p · ~ x − Et ) dEd p. (2.36)The convolution of this propagator with itself, with E and ~ p as variables, isNOT finite. It can be calculated, however, by appeal to distributions’ theoryusing the relation ^ f ∗ ^ g = ( ~ ) F ( fg ) . (2.37)This is so because divergences in the convolution of two phase space functionsderive from multiplication of distributions possessing singularities at the same configuration-space point. Keeping in mind that ∆ ( ~ x, t ) = (cid:16) m2πi ~ (cid:17) t − + e im ~ x2 ~ t , (2.38)72.37) yields ( ~ ) (cid:0) ^ ∆ η ( ~ p, E ) ∗ ^ ∆ η ( ~ p, E ) (cid:1) = Z (cid:16) m2πi ~ (cid:17) t − + e im ~ x2 ~ t e − i ~ ( ~ p · ~ x − Et ) dtd x. (2.39)The spatial integral is Z e im ~ x2 ~ t e − i ~ ( ~ p · ~ x ) d x = π ( i ~ t ) m e − i ~ p2t4 ~ m , (2.40)so that the convolution becomes ^ ∆ η ( ~ p, E ) ∗ ^ ∆ η ( ~ p, E ) = ( ~ ) (cid:16) miπ ~ (cid:17) Z t − + e i ~ ( E − ~ p24m ) t dt. (2.41)Using the result below (see [14]) F [ x λ + ] = ie iπλ2 Γ ( λ + )( k + i0 ) − λ − , (2.42)one finds ^ ∆ η ( ~ p, E ) ∗ ^ ∆ η ( ~ p, E ) = ~ m (cid:18) E − ~ p + i0 (cid:19) . (2.43) The classical FT associated to the q-Klein-Gordon equation was developedin [24]. Here we tackle the quantum version, whose action is S = Z (cid:8) ∂ µ φ ( x µ ) ∂ µ ψ ( x µ ) + ∂ µ φ † ( x µ ) ∂ µ ψ † ( x µ )− qm (cid:2) φ − ( x µ ) ψ ( x µ ) + φ † − ( x µ ) ψ † ( x µ ) (cid:3) (cid:9) d x. (3.1)This theory is 1) adequate for very energetic (TeVs) q-particles, according toCERN-Alice experiments, and 2) non re-normalizable for any q > 1 . Thus,it cannot be dealt with neither with dimensional regularization nor withdifferential one. A way out is provided by the ultradistributions’ convolutionof Bollini and Rocca [15, 16, 17, 18]. Ultradistribuions provides a generalformalism to treat non-renormalizable theories and gives in the configurationspace a general product in a ring with zero divisors (a product of distributions8f exponential type). For example we can treat cases with q ≥ as wewill do later.The concomitant theory is tractable here for weak fields and for A) q ∼ or B) particular q-values. We analyze first the case q ∼ , associated toenergies smaller that 1 TeV. We can thus write qm φ − = qm φ + ( q − ) m φ ln φ. (3.2)Since the field is weak we have φ ≃ I + ( q − ) η, (3.3)ln φ ≃ ( q − ) η, (3.4)Using (3.2), (3.3), and (3.4) the field’s action becomes S = ( q − ) Z (cid:8) ∂ µ η ( x µ ) ∂ µ ψ ( x µ ) + ∂ µ η † ( x µ ) ∂ µ ψ † ( x µ )−( − ) m (cid:2) η ( x µ ) ψ ( x µ ) + η † ( x µ ) ψ † ( x µ ) (cid:3) (cid:9) d x, (3.5)where we employed Z η ( x µ ) d x = Z ψ ( x µ ) d x = (3.6)Defining − = = ( − ) m . (3.7)one has S = ( q − ) Z (cid:8) ∂ µ η ( x µ ) ∂ µ ψ ( x µ ) + ∂ µ η † ( x µ ) ∂ µ ψ † ( x µ )− µ (cid:2) η ( x µ ) ψ ( x µ ) + η † ( x µ ) ψ † ( x µ ) (cid:3) (cid:9) d x. (3.8) The low energy field is just the usual Klein-Gordon one!
For the fields we have η ( x µ ) = ( ) Z " a ( ~ k ) √
2ω e − ik µ x µ + b † ( ~ k ) √
2ω e ik µ x µ d k, (3.9) ψ ( x µ ) = ( ) Z " c ( ~ k ) √
2ω e − ik µ x µ + d † ( ~ k ) √
2ω e ik µ x µ d k, (3.10)9here k = ω = q ~ k + µ . Field quantization proceeds then along familiar lines: [ a ( ~ k ) , a † ( ~ k ′ )] = [ b ( ~ k ) , b † ( ~ k ′ )] = [ c ( ~ k ) , c † ( ~ k ′ )] =[ d ( ~ k ) , d † ( ~ k ′ )] = δ ( ~ k − ~ k ′ ) . (3.11)For − = , i.e., q = , the low energy theory is one for a null mass field S = − Z (cid:2) ∂ µ η ( x µ ) ∂ µ ψ ( x µ ) + ∂ µ η † ( x µ ) ∂ µ ψ † ( x µ ) (cid:3) d x, (3.12)where k = ω = | ~ k | .We tackle now the q-KG theory for an integer n such that − = n , for m small, where the action is S = Z (cid:8) ∂ µ φ ( x µ ) ∂ µ ψ ( x µ ) + ∂ µ φ † ( x µ ) ∂ µ ψ † ( x µ )− n +
12 m (cid:2) φ n ( x µ ) ψ ( x µ ) + φ n † ( x µ ) ψ † ( x µ ) (cid:3) (cid:13) d x. (3.13)Now we define i) the free action S and ii) that corresponding to the inter-action S I as S = Z (cid:2) ∂ µ φ ( x µ ) ∂ µ ψ ( x µ ) + ∂ µ φ † ( x µ ) ∂ µ ψ † ( x µ ) (cid:3) d x, (3.14) S I = − n +
12 m Z (cid:2) φ n ( x µ ) ψ ( x µ ) + φ † n ( x µ ) ψ † ( x µ ) (cid:3) d x. (3.15)The fields in the interaction representation satisfy the equations of motionfor free fields, corresponding to the action S . This is to satisfy the usualmassless Klein-Gordon equation. As a consequence, we can cast the fields φ and ψ in the fashion φ ( x µ ) = ( ) Z " a ( ~ k ) √
2ω e ik µ x µ + b † ( ~ k ) √
2ω e − ik µ x µ d k, (3.16) ψ ( x µ ) = ( ) Z " c ( ~ k ) √
2ω e ik µ x µ + d † ( ~ k ) √
2ω e − ik µ x µ d k, (3.17)10here k = ω = | ~ k | The quantification of these two fields is i) immediatelytractable and ii) the usual one, given by [ a ( ~ k ) , a † ( ~ k ′ )] = [ b ( ~ k ) , b † ( ~ k ′ )] = [ c ( ~ k ) , c † ( ~ k ′ )] =[ d ( ~ k ) , d † ( ~ k ′ )] = δ ( ~ k − ~ k ′ ) . (3.18)The naked propagator corresponding to both fields is the customary one, andit is just the Feynman propagator for massless fields ∆ ( k µ ) = ik + i0 (3.19)where k = k − ~ k . The dressed propagator, which takes into account theinteraction, is given by ∆ ( k µ ) = ik + i0 − iΣ ( k µ ) , (3.20)where Σ ( k µ ) is the self-energy. Let us calculate the self-energy for the field φ at second order in perturbation theory, for which the only non vanishingdiagram corresponds to the convolution of n − propagators for the field φ and one propagator for the field ψ . All remaining diagrams are null.(this is easily demonstrated using the regularization of Guelfand for integralscontaining powers of x [14]). Therefore, we have for the self-energy theexpression Σ ( k µ ) = ( n + ) m (cid:18) ik + i0 ∗ ik + i0 ∗ ik + i0 · · · ∗ ik + i0 (cid:19) . (3.21)The convolution of n Feynman’s propagators of zero mass is calculated di-rectly using the theory of convolution of Ultradistributions [15]-[18]. Here,we just give the result, that turns out to be rather simple. A detailed demon-stration lies beyond this paper’s scope. We arrive at ik + i0 ∗ ik + i0 ∗ ik + i0 · · · ∗ ik + i0 = iπ ( n − ) k ( n − ) Γ ( n ) Γ ( n − ) (cid:2) ln ( k + i0 ) + ( ) − λ ( n − ) − λ ( n ) (cid:3) , (3.22)11here λ ( z ) = d ln Γ ( z ) dz . The self-energy is then Σ ( k µ ) = ( n + ) m ( n − ) k ( n − ) Γ ( n ) Γ ( n − ) (cid:2) ln ( k + i0 ) + ( ) − λ ( n − ) − λ ( n ) (cid:3) (3.23)For both fields φ and ψ , the self-energy and the dressed propagator coincideup to second order.Note that the current of probability is given by J µ = i4m [ ψ∂ µ φ − φ∂ µ ψ + φ † ∂ µ ψ † − ψ † ∂ µ φ † ] . (3.24)and it is verified that ∂ µ J µ = (3.25)This implies that the fields defined in the representation of interaction arephysical fields. We have here obtained some results that may be regarded as interesting.1) We developed the CFT for the particular q-SE advanced in [12].2) For this CFT we showed that the customary dispersion relations apply.We also introduced the physical fields i.e., those that the probability currentis conserved. The physical states are introduced via analogy with bosonicstring theory.3) We developed the QFT associated to the q-SE of [12]. For weak fields,this q-QFT coincides with the ordinary SE-QFT. This result confirms ourNuclear Physics A results. 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