aa r X i v : . [ phy s i c s . g e n - ph ] A ug Field Theory with Fourth-order Differential Equations
Rui-Cheng LI ∗ School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
Abstract
We introduce a new class of higgs type complex-valued scalar fields U withFeynman propagator ∼ /p and consider the matching to the traditional fieldswith propagator ∼ /p in the viewpoint of effective potentials at tree level. Withsome particular postulations on the convergence and the causality, there are a wealthof potential forms generated by the fields U , such as the linear, logarithmic, andCoulomb potentials, which might serve as sources of effects such as the confinement,dark energy, dark matter, electromagnetism and gravitation. Moreover, in somelimit cases, we get some deductions, such as: a nonlinear Klein-Gordon equation, alinear QED, a mass spectrum with generation structure and a seesaw mechanism ongauge symmetry and flavor symmetry; and, the propagator ∼ /p would providea possible way to construct a renormalizable gravitation theory and to solve thenon-perturbative problems. Key words linear potential, confinement, gravitation, dark energy
PACS Numbers ∗ [email protected] ontents ∂∂U ) term for kinetics term . . . . . . . . . . . . . . . . . . . . . 42.4 U is a kind of higgs-type field! . . . . . . . . . . . . . . . . . . . . . . . . . 5 U field . . . . . . . . . . . . . . . . . . . . . 63.2 The propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 h U i . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.2 Field U out a nutshell: generation of nonlinear Klein-Gordan equation . . . 135.3 The constraint U + U = h U i to a spontaneous breaking U (1) symmetry 145.3.1 U as a group element: the generation of gauge field A µ . . . . . . . 145.3.2 Multi-vacuum structure for sine-Gordon type vector field A µ . . . . 155.3.3 Duality between matter fields and media fields: from non-perturbativeto perturbative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 i Introduction
As a very successful theory, the gauge field theory with the gauge invariance principlecould be used to solve a huge part of questions for people. Certainly, there are somechallenges to the gauge theory: the extension for methods of application such as the onesfor non-perturbative problems, the extension for new phenomenons such as the ones fornew particles and new interactions, with an inevitable old topic about the unification andrenormalization.It’s just the linear potential from the non-perturbative results in lattice gauge theory [1]that motivated us to consider a fourth order differential equations. The motivation chainis: on the level of effective theories, we want to know what would be new in a theorywhich can generate a linear potential; mathematically, a straightforward way to constructlinear potential would be an introduction to Feynman propagator form of 1 /p , related tothe higher order differential equations; and, in the viewpoint of the superficial degree ofdivergence, the new Feynman propagator ∼ /p associated the higher order differentialequations might provide us a construction to a renormalizable gravitation. So, it wouldbe significant to investigate models in the higher order differential equations formalism, incombining with the treatment on puzzles on the redundant unphysical degrees of freedom(d.o.f). For simplicity, we will concentrate our studies on the pro forma feasibility of themodel in a view of effective potentials at tree level.The remainder of this paper is organized as follows: Sect. 2 is for the Lagrangian con-struction for a linear potential; Sect. 3 is for the kinetics and the propagators from theLagrangian; Sect. 4 is for the effective potentials generated from the Lagrangian, espe-cially for the linear, Coulomb and gravitational potentials; Sect. 5 is for some interestingdeductions uniquely occurring in our theory for some limit cases; Sect. 6 is for interpretingthe causality in our theory; and Sect. 7, the final section, is for our conclusions. We can get the classic non-relativistic (NR) potential forms from the amplitudes of thetree-level 2 → ⊗ (inner-line propagator) ⊗ (vertex) ⇔ V (1)where the l.h.s is a part of the amplitude for a tree-level Feynman diagram, and ther.h.s is the classic potential. So, conversely, we can build theories for potentials with adefinite form through the tree-level-correspondence, provided that the theories are per-turbatively computable. For example, if there were neither momentums nor coordinatesin the Feynman rules of vertices, we would extract different potentials with differentinner-line propagators, such as:linear potential ↔ p , Coulomb potential ↔ p , short-distance potential ↔ p α , with ∞ < α < . (2)1 .2 Lagrangian Firstly, we take a complex-valued scalar field U , a Dirac field ψ (and ¯ ψ ) as the physicalfield degree of freedom(d.o.f) , which have the transformation law under a global U (1)group element V as U → V U V − = U , ψ → V ψ , ¯ ψ → ¯ ψV − . (3)Secondly, in the method mentioned in Section 2.1, for a theory with a propagator form ∼ p for U , we write the Lagrangian of { U, ψ, ¯ ψ } as L = L U + L ψ + L I , (4)where the term L U = T r (cid:8) − ∂ µ ∂ ν U † ∂ µ ∂ ν U − Λ U [( U + U † ) + i ( U − U † )]+ m U U † U − λ U Λ U U † U U † U (cid:9) (5)is purely of the complex-valued scalar field U , the term L ψ = ¯ ψ ( i∂/ − m ψ ) ψ (6)is purely of the matter field ψ , and the term L I = − α Λ Q α ¯ ψ (cid:8) [( U + U † ) + i ( U − U † )] (cid:9) ψ − βQ β ¯ ψ (cid:8) σ µν ∂ ν [( U + U † ) + i ( U − U † )] (cid:9) γ µ ψ − ξ M Q ξ ¯ ψ (cid:8) σ µν ∂ µ [( U + U † ) + i ( U − U † )] (cid:9) ( i ←→ ∂ ν ψ ) (7)is the invariant interaction term of ψ coupled to U under the transformations in (3) and theLorentz transformation. The application of σ µν in the β term is to ensure a real-valued ef-fective coupling in the Feynman rule language, by recalling the reduction of − iσ νµ q µ → q ν .Thirdly, we give some postulations as the illustrations of the variables in the La-grangian of (4) as below.1. Λ U is a constant of the dimension of mass, m U is the mass of field U , and λ U is adimensionless constant; m ψ is the mass of field ψ .2. For the real-valued coefficients, there is α, β, ξ >
0. Particularly, if there is α = β = ξ , there is a kind of symmetry between the intrinsic charges and the momentumsof the matter fields ψ , which seems like a kind of realization of the supersymmetry.3. For the parameters Λ and M , referring to Wilson’s scheme for renormalization, forthe interaction Lagrangian terms we can propose the postulations as:(i) each U (not ∂U ) is tied with one infrared (I.R.) energy scale Λ; Discussions for the vector field U µ , the tensor field U µν , and the massive { U, U µ } have also beenfinished by the author, see Ref. [3]. If we define U = U + iU , then the global U (1) symmetry, or a global SO (2) symmetry, is definedbetween U and U . D >
4) are suppressed by a ultraviolet (U.V.)energy scale M .For example, if we plan to construct a QED, a QCD or a gravitation theory with the U field, then the variable Λ and M might be respectively set asΛ = µ IR ≃ { , Λ QCD , } , M = µ UV ≃ { µ EW , µ GUT , µ
P lank } , (8)where µ IR is the I.R. boundary energy scales, i.e., { value ≃
0, the QCD scale Λ
QCD ≃ M eV , value ≃ } , and M is the U.V. boundary for the theory, i.e., { the electroweak(EW) scale µ EW ∼ GeV , the grand unification theory (GUT) scale µ GUT , the Plankscale µ P lank } , for a QED,a QCD and a gravitation theory, respectively.4. The variables Q { α,β,ξ } can be seemed as a kind of reconstructed charges (RC), andthey are defined as Q { α,β,ξ } ≡ Y Q { α,β,ξ } , Y = ± , (9)where Y is the generator of the global U (1) group with eigenvalues ±
1, and Q α is a gen-erator of some other global group (such as the electromagnetic U (1) group) correspondingto the current J α , with the definitions Q α ≡ , Y ; J α ≡ ¯ ψψ ; (10) Q β ≡ T Q ; J β ≡ ¯ ψγ µ ψ ; (11) Q ξ ≡ Y ; J ξ ≡ ¯ ψi ←→ ∂/ ψ, (12)where Q α = 1 for neutral U (e.g. U for mediating a QED theory), Q α = Y for charged U (e.g. U for mediating a QCD theory); T Q ≡ T QED , T aQCD , ... , is either the generator ofthe QED U (1) group for constructing a QED theory with U , or one of the the generatorof QCD SU (3) group for constructing a QCD theory with U , etc.Furthermore, if we define a kind of effective media field as( A I ) α ≡ − α Λ Q α · [( U + U † ) + i ( U − U † )] , (13)[( A I ) β ] µ ≡ − β Q α · σ µν ∂ ν [( U + U † ) + i ( U − U † )] , (14)then the interaction Lagrangian terms in (7) can be expressed as L I ≡ L RC = ( A I ) · J · Y. (15)5. How to determine the value of Y and Q α ? Here we define: if the momentum of U flows “in” to the ¯ ψU ψ vertex, then the charge at this vertex is Y = +1, motivated byan imagination that the effective mass of ψ would become bigger by “eating” a nonzerovacuum expectation value h U i ; on the contrary, if the momentum of U flows “out” of the¯ ψU ψ vertex, then the charge at this vertex is Y = −
1. Similarly for the Q α , Q β and Q ξ ,e.g.:(i) for Q α : in the case of a charged U for a QCD theory, in every physically allowedprocess, if the Q α charge of U flows “in” to the ¯ ψU ψ vertex, then the Q α charge varia-tion for the “current” J α ≡ ¯ ψ i ψ j (with i, j the color indices) at this vertex is Q α = +1,the same as the value of Y ; on the contrary, if the Q α charge of U flows “out” of the¯ ψU ψ vertex, then the Q α charge variation for the “current” J α ≡ ¯ ψ i ψ j at this vertex is Q α = −
1, the same as the value of Y ; in the case of a neutral U for a QED theory, the3 α charge variation for the “current” J α ≡ ¯ ψψ at both vertices are defined to be always1; (ii) for Q β : even in the case of a neutral U for a QED theory, the Q β charge varia-tion for the “electromagnetic current” J β ≡ ¯ ψγ µ ψ is not 1, but to be the QED “charge” T QED ≡ Q
QED ;(iii) for Q ξ : even in the case of a neutral U for a gravitation theory, the Q ξ chargevariation for the “momentum current” J ξ ≡ ¯ ψi ←→ ∂/ ψ is not 1, but to be Y ; etc.6. To ensure the renormalizability, we need an extra postulation : all divergencescan be removed by introducing cutoff for the amplitudes or the phase-space parameters.More detail have been discussed in Ref. [3]. ( ∂∂U ) term for kinetics term The traditional kinetic term ( ∂U ) and U † ∂∂U will not appear in our model, which is like the case that a term U † ∂U will not appear inthe kinetic term of a Klein-Gordon field; this might be related to a kind of generalized“charge” symmetry. So, our theory with high-order derivatives is different from the onesdiscussed by Ostrogradski (or a quantized version by Pais, Uhlenbeck) [4] or the so-called f ( R ) theories discussed in general relativity formalism [5].For convenience, we would call the model for U defined with the ( ∂∂U ) term forkinetics term as a “ P4 type ”, and the traditional model for U defined with the ( ∂U ) term for kinetics term as a “ P2 type ”.It might be helpful for us to more easily understand the double partial term ( ∂∂U ) for the kinetics term, if we understand our U field as a classic continuum medium field.For the detail, for the continuum medium field φ we have the continuity equation ∂ µ ∂ ν T µν = 0 , (16)with the energy-momentum tensor defined as T µν = ( ρ + p ) u µ u ν + pg µν = ∂ L ∂ ( ∂ ν φ α ) ∂ µ φ α − g µν L = ( ∂ µ φ † ∂ ν φ + ∂ ν φ † ∂ µ φ ) − g µν ( ∂ α φ † ∂ α φ − m φ † φ )= φ † h ←− ∂ µ ∂ ν φ + ←− ∂ ν ∂ µ − g µν ( ←− ∂ α ∂ α − m ) i φ = φ † (cid:2) ( i∂ µ i∂ ν φ + i∂ ν i∂ µ ) − g µν ( i∂ α i∂ α − m ) (cid:3) φ. (17)Formally, to fully describe a field φ , one might need the ∂∂ · ∂∂ operator acting on thefield.Moreover, we can write the E.O.M in another form,ˆ p U ( x ) = [ˆ p Φ( x )] = [ˆ p e Φ( x )] · [ˆ p Φ( x )] , (18)with the correspondence for e Φ to Φ here is just like a generalized version of the case thatthe anti-particles ¯ ψ associated with the particles ψ , which also arised from the treatment4hat the Dirac equation was formally from the square root of the Klein-Gordon equation.Besides, we can see, if the E.O.M is not the form ˆ p U = m U , then that might break ageneralized “charge” symmetry between Φ and e Φ. We can denote that asΦ( x ) ∼ h ¯ ψψ i ⇒ K-G eq. = [ Dirac eq. ] , (19) U ( x ) ∼ h e ΦΦ i ⇒ U-eq. = [ K-G eq. ] . (20)Then we can have the new E.O.Mˆ p Φ = m U Φ ⇒ Φ = c e ip · x + c e − ip · x (21)for the ordinary physical d.o.f, andˆ p e Φ = − m U e Φ , (tachyon/higgs) (22) − ˆ p e Φ = m U e Φ , (phantom) (23) ⇒ Φ = d e p · x + d e − p · x omit divergent terms −−−−−−−−−−−→→ d (cid:2) e p · x · θ ( − p · x ) + e − p · x · θ ( p · x ) (cid:3) , ( p · x = 0) , (24)for the so-called unphysical d.o.f (with the θ function being the step function): thetachyons in (22), with an imaginary number valued mass [6]; and the phantoms in (23),with a negative kinetic energy [7], respetively. The sign of the action corresponding tothe E.O.Ms in (22) and (23) are different, which is not negligible [8].Although there exist unphysical and acausal solutions in addition to the two physicald.o.f for differential equations with orders higher than 2 in the classic mechanics case, we can avoid this trouble by treating these solutions as effects of hidden unphysical newd.o.f (which are existent but can’t be directly measured for some reasons) beyond thestandard model (SM) in particle physics; this is to discussed in the following Section 2.4.We will revisit this topic in Section 6, and we want to propose that unphysical d.o.f doesnot necessarily mean acausality. U is a kind of higgs-type field! The self-interaction potential of field U is V ( U ) ≡ − m U U † U + λ U Λ U U † U U † U, (25)so, according to the minus sign in the mass term, U is a kind of higgs-type field. And,for convenience, in all this article for allowed cases we set h U i = 1 . (26)But we should remind ourselves that h U i could be very large even when the energy scaleis very low.For a higgs field U with a potential form in (25) plotted as the line-“b” in Fig.-1-(1),besides of the angular component U θ as the conventional field (the Goldstone boson),there is also a radial-direction component 0 ≤ U r ≤ + ∞ . Here, the most important pointis, how to understand the U r ? The acausality discussed in Ref. [9] only occurs in the classic mechanics case and can be removedin the formalism of quantum mechanics through the uncertainty principle by treating all the observablevariables as operators.
51) (2)Figure 1: Self-interaction potentials for the field U and A .For a potential V ( U ) of the form as the line-“a” in Fig.-1-(1), which is defined only for0 ≤ | U | ≤ | U | < ∞ field configurations, we can not only treat theradial-direction component U r as a stable (physical) fluctuation around the stable vacuum | U | = 1 (minimum of the potential V ( U )), but also treat U r as a 0 ≤ U r ≤ | U | = 0 maintained by the rebound from the potential barrier. Similarly,for a potential V ( U ) of the form as the line-“b” in Fig.-1-(1), we can also understand theradial-direction component 0 ≤ U r ≤ + ∞ in two viewpoints: U r is a stable (physical)field d.o.f U higgsr oscillating around the stable vacuum | U | = 1 (minimum of the potential V ( U )), which could be seemed as the “traditional” P2 type excitation of “higgs particle”;or, U P r is an unstable (unphysical) field d.o.f oscillating around the unstable vacuum | U | = 0 (local maximum of the potential V ( U )), which would “decay/collapse” as whatwould happen in the more extreme two cases plotted as the line-“c” or line-“d” in Fig.-1-(1)).However, we will just take the unstable (unphysical) U P r d.o.f as the real componentin our “untraditional” P4 type U field, with the purpose to design the U field to differfrom the “traditional” P2 type field. Thus, from now on, we need not give too manyquery to the sign of the mass term in (5) any more. As discussed in Section 2.3, we cansay: U is a kind of higgs-type field, and U does have a nonzero VEV, however, the U fieldwith E.O.M. ˆ p U = m U is really designed to be neither a traditional higgs field withE.O.M. ˆ p U = − m U nor a phantom with E.O.M. − ˆ p U = m U , see (22).In a word, it should be emphasized that the choice for the sign of the mass term isvery important and crucial for our following work. U field By the Euler-Lagrange equation [10] ∂ L U ∂U − ∂ µ ∂ L U ∂ ( ∂ µ U ) + ∂ µ ∂ ν ∂ L U ∂ ( ∂ µ ∂ ν U ) = 0 , (27)from (5) we can get the equation of motion(E.O.M) of free field U, − ∂ µ ∂ ν ∂ µ ∂ ν U = − m U U + Λ U (28) ⇔ − ˆ p U = − m U U + Λ U , ˆ p µ = i∂ µ , (29)6nd the dynamical E.O.M for U , as − ∂ U = − m U U + Λ U + αQ Λ ¯ ψψ + ... . (30) By inserting the “correlation function”, i.e., one version of the definitions of propagatorof U , D F ( x − y ) ≡ h Ω | ˆT U ( x ) U ( y ) | Ω i = θ ( x − y ) h Ω | U ( x ) U ( y ) | Ω i + θ ( y − x ) h Ω | U ( y ) U ( x ) | Ω i (31)into the E.O.M, where ˆT is the time-ordering operator, | Ω i is the vacuum state, we canverify − ( ∂ − m ) x D F ( x − y ) ≡ ( ∂ − m ) x h Ω | ˆT U ( x ) U ( y ) | Ω i = + iδ (4) ( x − y ) . (32)That means, D F ( x − y ) is really the “Green function”, i.e., the other version of the defi-nitions of propagator of U .By setting Λ U = 0, from (32) or its corresponding form in the momentum space − ( p − m U ) e D F ( p ) = i , (33)we can get the Feynman propagator for m U = 0 case in the momentum space, as e D F ( p ) = − ip − m U + iǫ = − i ( p + m U − iǫ )( p − m U + iǫ ) , (Λ U = 0, m U = 0) , (34)or, for m U = 0 case D F ( U ) = − ip + iǫ , (Λ U = 0, m U = 0) . (35)So, the minus sign before the ˆ p operator in the E.O.M (29,30,32,33) is very crucial,which represents the sign of the mass term in Lagrangian, and, without this “ −
1” factor,everything will be different! After all, the U here isn’t the traditional scalar field, aswe said in Section 2.4. Besides, the position and residue of a pole in the propagator iscrucial for the calculation results of the amplitudes. In this work, we will only considerthe m U = 0 case, and the m U = 0 case has been discussed in Ref. [3]. At the beginning, we set the variables for the particles in the scattering processesshown in Fig. 2, as below: p = ( m, p ) , p = ( m, p ) , (36) p ′ = ( m, p ′ ) , p ′ = ( m, p ′ ) . (37)7 ′ p p ′ p qp ′ p p p ′ q ( a ) ( b ) Figure 2: The Feynman diagrams for the leading order tree level processes, with (a)mediated by a U and (b) mediated by a photon A µ .In the non-relativistic approximation, q = 0 (which is also called on-shell approximation),we have the relations for kinetics variables as q = p − p ′ ⇒ q = ( p − p ′ ) ======= q =0 −| q | = −| p − p ′ | , (38)and ¯ u s ′ ( p ′ ) u s ( p ) = 2 mδ ss ′ , ¯ u s ′ ( p ′ ) γ µ u s ( p ) (NR limit) ======= v µ mδ ss ′ . (39)Besides, In the non-relativistic limit, we need not consider the identical particle effects,that is, we need not consider the u channel of the Feynman diagrams in a scattering processnow. Now, for the interaction term L αβ = − α Λ Q α ¯ ψ (cid:8) [( U + U † ) + i ( U − U † )] (cid:9) ψ − βQ β ¯ ψ (cid:8) σ µν ∂ ν [( U + U † ) + i ( U − U † )] (cid:9) γ µ ψ, (40)which was extracted from the total interaction Lagrangian (7), by defining the couplings α , ≡ α ( Q α ) , , β , = β ( Q β ) , , (41)and by using iσ µν q ν = − q µ , γ µ → v µ from (39), we can write the corresponding amplitudefor Fig. 2-(a), as i M a = ¯ u s ′ i [ − α Λ − β σ µν ( iq ν ) γ µ ] u s · − iq · ¯ u r ′ i [ − α Λ − β σ αβ ( − iq β ) · δ µα γ α ] u r = ¯ u s ′ i [ − α Λ + β q µ γ µ ] u s · − iq · ¯ u r ′ i [ − α Λ − β q α · δ µα γ α ] u r = ¯ u s ′ [ − iα Λ] u s · − iq · ¯ u r ′ [ − iα Λ] u r +[ α Λ β ( q · v ) − α Λ β ( q · v )] · − iq · ¯ u s ′ u s ¯ u r ′ u r +¯ u s ′ [ iβ γ µ ] u s · − ig µα q · ¯ u r ′ [ − iβ γ α ] u r , (42) For simplicity, here we can only consider the contributions from U , and, for the contributions from U , the result just need a double. δ µα are not controlled by the Einstein summation convention, andwe have taken the replacement q µ q α δ µα → q g µα . (43)The use of the δ µα in (42) could be understood by this reason: as there is only the single U field exchanged in a 2 → µ -component of the momentum q µ of U is absorbed into one vertex in the Feynman diagram, there must be the same µ -component of the momentum q µ is emitted out from the other vertex in the Feynmandiagram! Thus, by treating ∂ µ U ∼ A µ as an P2 type effective media field as in (14), forwhich there is a propagator with the form of − ig µα q , we can see, the last term in (42) isjust of the form of an amplitude for a scattering process corresponding to the Coulombpotential in QED, as shown in Fig.2-(b).In the NR limit of q = 0, with the definitions q · v ≡ λ | q | , q · v ≡ λ | q | , (44)and the approximation γ µ γ α g µα → γ γ g = 1, we can continue to get i M a = − i (cid:26) − α α Λ | q | + ( α β λ − α β λ )Λ | q | − β β | q | (cid:27) · ¯ u s ′ u s · ¯ u r ′ u r , (45)The amplitude i M should be compared with the Born approximation to the scatter-ing amplitude in non-relativistic quantum mechanics, written in terms of the potentialfunction V ( x ): [2] i M ∼ NR h p ′ | iT | p i NR = − i e V ( q )(2 π ) δ ( E p ′ − E p ) , q = p − p ′ , (46)with p = η p − η p , p ′ = η p ′ − η p ′ , η i = m i m + m , i = 1 , . (47)By dealing with the kinetics factors as 2 mδ ss ′ → δ ss ′ and (2 π ) δ ( E p ′ − E p ) →
1, we canhave e V ( q ) = − α α Λ | q | + ( α β λ − α β λ )Λ | q | − β β | q | , (48)and the inverse Fourier transformation V ( x ) = F − [ e V ( q )] . (49)Then, we can get the potential V ( r ) = + α α Λ π r − ( α β λ − α β λ )Λ2 π (log rr + γ E −
1) + − β β πr , (50)with r = 1 GeV − put by hand to balance the dimension, and γ E the Euler constant.Moreover, by applying (9,10,11, 41) to get (for QED:) α α = α ( Q α ) ( Q α ) = α ( Y · ( Y · = − α , (51) − β β = − β ( Q β ) ( Q β ) = − β ( Y Q β ) ( Y Q β ) = − β ( Y Q QED )( Y Q QED ) = β Q QED Q QED , (52) α β = αβ ( Q α ) ( Q β ) = αβ ( Y · ( Y Q
QED ) = − αβ Q QED , (53) α β = αβ ( Q α ) ( Q β ) = αβ ( Y · ( Y Q
QED ) = − αβ Q QED , (54) It is very important to set Y = − Y = +1 to match the Feynman rules used in Eq. (42)! Q QED just the electric charge in QED, and(for QCD:) α α = α ( Q α ) ( Q α ) = α ( Y · Y ) ( Y · Y ) = α , (55) − β β = − β Q QCD Q QCD , (56) α β = αβ ( Y · Y ) ( Y Q
QCD ) = αβ ( Y Q QCD ) = αβ Q QCD , (57) α β = αβ ( Y · Y ) ( Y Q
QCD ) = αβ ( Y Q QCD ) = − αβ Q QCD , (58)with Q QCD just the color charge in QCD, by combining with (41,51,52,53,54), the potentialin (50) will become V ( r ) QED = − α Λ QED π · r + αβ Λ QED ( Q QED λ − Q QED λ )2 π · log rr + β Q QED Q QED πr , (59) V ( r ) QCD = α Λ QCD π · r + αβ Λ QCD ( Q QCD λ + Q QCD λ )2 π · log rr − β Q QCD Q QCD πr , (60)By recalling that we have performed the derivations in the formalism of a collision pro-cess in the center-of-mass frame, that is to say, p = − p , by combining the on-shellapproximation | p | = | p ′ | , we can indeed determine the relations q · ( p + p ) = ( m λ + m λ ) | q | = 0 , q · ( p − p ) = ( m λ − m λ ) | q | > , ( q · p )( q · p ) = m m λ λ | q | < . (61)So, for the case of m = m , we will have λ − λ > λ + λ = 0. As in (42), wecan see again, the last term in (59) or (60) is coincidentally for the Coulomb interactionin QED or QCD, respectively!Besides, there is a linear potential and a logarithmic potential in both (59) and (60).In (59), since the infrared energy scale boundary Λ QED for the QED is about zero, thelinear potential and the logarithmic potential could be negligible; however, in some cos-mological experiments, the linear and the logarithmic term might give corrections to theelectromagnetic observables, such as: a spatial variation of the electromagnetic fine struc-ture constant [11], or a kind of electromagnetic red-shift coupled with the gravitationalred-shift. In(60), since the infrared energy scale boundary Λ
QCD for the QCD is about200
M eV , the linear potential could be significant to serve as the major part of the con-finement in QCD, while the logarithmic potential could serve as a minor part of theconfinement.
Now we consider the interaction terms, L αξ = − α Λ Q α ¯ ψ (cid:8) [( U + U † ) + i ( U − U † )] (cid:9) ψ − ξ M Q ξ ¯ ψ (cid:8) σ µν ∂ µ [( U + U † ) + i ( U − U † )] (cid:9) ( i ←→ ∂ ν ψ ) , (62)10hich was extracted from the total interaction Lagrangian (7). By defining the couplingsas in (41) α , ≡ α ( Q α ) , , ξ , ≡ ξ ( Q ξ ) , (63)and by using iσ µν q ν = − q µ , q ν q β δ νβ → q g νβ as in (42,43), we can write the correspondingamplitude for Fig. 2-(a), as i M = ¯ u s ′ i (cid:26) − α Λ − ξ M σ µν · ( iq µ ) · [ i · i ( p + p ′ ) ν ] (cid:27) u s · − iq · ¯ u r ′ i (cid:26) − α Λ − ξ M σ αβ · ( − iq α ) · δ νβ · (cid:2) i · i ( p + p ′ ) β (cid:3)(cid:27) u r = − iq · (cid:26) − α α Λ + Λ M q · [ α ξ ( p + p ′ ) − α ξ ( p + p ′ )]+ ξ ξ M · q ν q β δ νβ · ( p + p ′ ) ν ( p + p ′ ) β (cid:27) · ¯ u s ′ u s · ¯ u r ′ u r = − i · (cid:26) − α α Λ | q | − Λ M · (cid:20) (2 α ξ q · p − α ξ q · p ) | q | (cid:21) + Λ M · (cid:20) ( α ξ + α ξ ) | q | (cid:21) − ξ ξ M · p · p | q | + ξ ξ M · q · p − q · p ) | q | − ξ ξ M (cid:27) · mδ ss ′ mδ rr ′ , | q | > . (64)In the non-relativistic limit, with p , = m , v , , and the definitions q · v ≡ λ | q | , q · v ≡ λ | q | , (65)we can get q · p = m λ | q | , q · p = m λ | q | . (66)Thus the non-relativistic effective potential in the momentum space will be e V ( q ) = −M = − α α Λ | q | − Λ M · (cid:20) (2 α ξ m λ − α ξ m λ ) | q | (cid:21) + Λ M · (cid:20) ( α ξ + α ξ ) | q | (cid:21) − ξ ξ M · p · p | q | + ξ ξ M · m λ − m λ ) | q | − ξ ξ M , | q | > . (67)The last term in (67), − ξ ξ M , is effective to a Feynman rule of a vertex for a four-fermioncontact term, so we will drop it in the non-relativistic limit due to the probability con-servation law in the non-relativistic quantum mechanics formalism.Then, by performing the inverse Fourier transformation V ( x ) = F − [ e V ( q )], we canget the potential in the coordinate space (with | q | > θ ( | q | ),11 E the Euler constant) as V ( r ) = α α Λ π r − Λ(2 α ξ m λ − α ξ m λ ) M · (cid:20) − π (log rr + γ E − (cid:21) + Λ( α ξ + α ξ ) M · πr − ξ ξ p · p M · πr + 2 ξ ξ ( m λ − m λ ) M · π ir δ ( r ) , ( r > , (68)with r = 1 GeV − put by hand to balance the dimension. The last term of the δ ( r )function in (68) is from the | q | term in (67), and it could also be dropped due to r = 0.At last, with the values Y = − Y = +1, we have α ξ = αξ ( Q α ) ( Q ξ ) = αξ ( Y · ( Y · Y ) = − αξ, (69) α ξ = αξ ( Q α ) ( Q ξ ) = αξ ( Y · ( Y · Y ) = αξ, (70) ξ ξ = ξ ( Q ξ ) ( Q ξ ) = ξ ( Y · Y ) ( Y · Y ) = ξ , (71)by combining with α α = − α in (51) and m λ + m λ = 0 in (61), we can get thepotential form V ( r ) = − α Λ π r − ξ p · p M · πr + αξ Λ( m λ + m λ ) π M · log rr , r > . (72)As expected, the linear potential also arises in (72) is the same as in (59), which could becorresponding to the dark energy effect (or the gravitational red-shift) and the inflationeffect in a Big-bang universe. And the second term in (72) is happily to be the Newton’sgravity form! Besides, a potential term with form of − v r included in the factor p · p = p p − p · p ≃ m p − v · m p − v + m | v | (73)with p = − p in the center-of-mass frame, can be treated as one of the source of thedark matter effects [12], which is just of a relativistic effects! Moreover, there will be anextra relativistic corrections from the spinor basis u s ( p ), by replacing m to p in (39), as¯ u s ′ ( p ′ ) u s ( p ) = 2 p δ ss ′ , ¯ u s ′ ( p ′ ) γ µ u s ( p ) (NR limit) ======= v µ mδ ss ′ . (74)The logarithmic term in (72) would also be treated as one of the source of the darkmatter effects [12] in the case of m λ + m λ >
0, which means some kinds of C parityasymmetry effects arise in addition to (61), i.e., α ξ m λ = α ξ m λ ; and, r shouldbe big enough so that the dark matter effects would be at least of the same order of theNewton’s gravity.In the sense of the superficial degree of divergence, the gravitational interaction termin (62) is renormalizable, so, a construction of a renormalizable gravitation theory mightbe practicable in our P4 type formalism. And, this P4 formalism might also be useful torenormalize the scalar QED or the chiral perturbative theory, etc.Besides, we want to point out that, for a N -body system, potential terms in (72)will be additive and they will be enlarged only by the factor ( N Q ) · ( N Q ), rather than( N ξ )( N m ) · ( N ξ )( N m ). 12 Induced theories in some limit cases h U i Now that U is a kind of higgs field, it should show its higgs-like property. According tothe higgs mechanism, with the interaction term α Λ ¯ ψU ψ in (7), the fermion (or similarly,the boson) matter fields will get a mass correction∆ m ∼ α Λ h U i . (75)For a very small ∆ m value, it might serve for the mass of very light particles as darkmatter candidates, or, instead of the axion [13], it might present a solution to the strongCP (naturalness) problem.If we set h U i G = L ≃ − GeV as the gauge symmetry breaking energy scale ofgravitation, with L ≃ l.y. corresponding to the size of the universe, and h U i EW ≃ GeV as the gauge symmetry breaking energy scale of electroweak interaction, we willget a lucky coincidence for the ratio of the magnitudes of Newton’s gravity force F G andthe Coulomb force F C , F G F C = (cid:20) G (cid:16) m e e (cid:17) e r (cid:21) / (cid:20) k e r (cid:21) ≃ − → h U i G h U i EW (76)where m e is the mass of electron, k ≃ × ( N · m · C − ) is the Coulomb constant(inSI unit). If this is true, we might say, the smallness of gravitation constant G comes fromits small VEV h U i G (or the huge size of the universe).Furthermore, if we set h U i T C as the gauge symmetry breaking energy scale of thetechnicolor (TC) interaction [8], and the ratio h U i T C h U i EW ≃ g s e ≃ . . = 100 (77)will give us a value of h U i T C ≃ GeV = 10
T eV for the typical energy scale of techni-color dynamics. U out a nutshell: generation of nonlinear Klein-Gordanequation Here we need the self-interaction term of U , which could be written as L I = − g U Λ U U ∂ µ U ∂ µ U + m U U . (78)For a pure U -field system, if its kinetic energy is very small, down to p ≪ Λ U Λ (or, inthe sense of de Broglie wavelength, we can say, the system is “out of a nutshell”), thenthe kinetic energy term could be dropped, then we can get a E.O.M for U according tothe Euler-Lagrangian equation, as g U Λ U ( ∂U ) − g U Λ U U ∂ U = m U U ⇒ ( ∂U ) − U ∂ U = m U g U Λ U U . (79)Apparently, that is a nonlinear 2nd-order differential equations, so, we just call it “non-linear Klein-Gordon equation”. Particularly, for a special case, h U i ≫ U − h U i (i.e., the13EV large and the fluctuation small) and h U i ≫ ∂U (i.e., the VEV large and the kineticenergy small), we can get the “linear” Klein-Gordon equation − ∂ U = m U g U h U i Λ U U , (80)and there should be the relation 2 g U h U i Λ U = m U . As said for (26), we should remindourselves that h U i could be very large even when the energy scale is very low!In a Lagrangian, there should be both the kinetic energy terms and the potentialenergy terms. However, there exists the freedom to choose which ones are the kineticenergy terms and which ones are the potential energy terms, that depends the choice ofthe d.o.f of the system. This is a kind of “kinetic-potential duality”. U + U = h U i to a spontaneous breaking U (1) symmetry U as a group element: the generation of gauge field A µ To a spontaneous breaking U (1) symmetry, if we take the constraint U + U = h U i ,there will be U = U + iU = σ ( x ) e − iφ ( x ) → h U i e − igφ ( x ) ≡ u, (81)that is, if we choose the unitary gauge condition σ = 0, U will become a group element.In (81), U and U are both P4 type field, and σ and φ are also both P4 type field; σ is purely unphysical field (i.e.,tachyon/instanton/phantom), while φ is physical field,as said in Sect. 2.4. Is the φ ( x ) really a detectable field? Mathematically to say, φ is aphase, and we can write U → u = h U i e − igφ ( x ) → h U i e − ig [ φ ( x )+ ǫn µ A µ ( x )+ ǫn µν A µν ( x )+ ... ] , (82)that means, the P4 type φ field can be generated by many different fields rather than onlyone field φ ( x ). If only the A µ ( x ) field is nonzero in (82), then, with¯ ψ ( U ∂/U † ) ψ → ¯ ψ ( u∂/u † ) ψ → e ¯ ψA/ψ, (83)as a 4-particle-coupling term becoming to a 3-particle-coupling term, we get the gaugeinteraction term, with β · h U i = e . (84)Now, instead of the d.o.f. of φ ( x ), there exists a connection field (gauge filed) A µ ( x ), in-duced by the Maurer-Cartan 1-form of u ( x ) field. Thus, the superficial gauge symmetry of the Lagrangian arises! We name the constraint U + U = h U i , A µ ( x ) = 0 , φ ( x ) = A µν ( x ) = ... = 0 (85)as “ Light Constraint ”, in the reason that it survive only the field A µ with the lightspeed after freezing the unphysical tachyon d.o.f. σ ( x ) in (81) with speed over the light.However, when both U and U are excited, the contribution of the massless U fieldincludes an effect of a massless gauge field A µ ( x ), see Fig. 2-(a). Now, as both the ¯ ψ∂U ψ term and the ¯ ψA µ ψ term can generate the Coulomb potential, we would like to ask, isthe gauge symmetry necessary? We will return this question in Sect. 6. It is reasonable, by reminding that a Dirac spinor field could even be formally constructed as thesquare root of a scalar field φ . .3.2 Multi-vacuum structure for sine-Gordon type vector field A µ
1. Multi-vacuum structure for A µ If we write U ( x ) = exp[ − igǫn µ A µ ( x )] = cos[ gǫn µ A µ ( x )] − i sin[ gǫn µ A µ ( x )] , (86)then the potential term V ( A ) ∼ U ( A ) + U † ( A ) = cos[( gǫ ) · A ] , (87)would mean that the dynamics for the field A µ is of a sine-Gordon type (or, a kind ofgeneralized higgs type vector), see Fig. 1-(2). Thus, there might be many excitations for A µ at different vacuums (or, VEVs), with heavy masses in the large g cases( gǫ ≃
1) andsmall masses in the small g cases.2. Mass spectrum with generation structureLike the mass correction in (75) from U , with the term ¯ ψA/ψ , the fermion (or similarly,the boson) fields can get a mass correction from A µ ,∆ m ∼ α Λ h A i ∼ α Λ (2 n + 1) πgǫ , n = 0 , , , ... . (88)where the number n might lead the fermion mass spectrum to a generation structure.Even for the same value of n , we can get the deductions below:a. if ∆ m is the mass differences between the current quarks and the constituentquarks, then, by setting g ∼ (2 n + 1) α Λ∆ m · ǫ O (Λ) ∼ O ( ǫ ) −−−−−−→ (2 n + 1) α ∆ m ∼ , (89)with ∆ m ∼ GeV and n = 0, we have α ∼ g ∼ .
01 for the E.W. interaction, then, ∆ m ∼ GeV , corresponding to thepossible heavy fermions.3. A seesaw mechanism for gauge symmetry and flavor symmetrySee Fig. 1-(2), with (87), for a vacuum at A = h A i i , the potential could be written as V ( A ≃ A i ) ≃ − gǫ ) ( A − A i ) + . . . , (90)which means the mass of the excitation A ′ = A − h A i i is of order ∼ m = gǫ . So, we canget the conclusions below:(1) when g → A ′ µ is nearly massless, so the gauge symmetry is restored;b. the VEV h A i i are of very different magnitudes, so, through (88), the fermion masseswould be also of very different magnitudes, including very heavy fermions; this is a kind As said for (26), we should remind ourselves that h U i could be very large even when the energy scaleis very low!
15f flavor symmetry breaking for fermions;(2) when g → ∞ ,a. A ′ µ is massive, with the diagonal elements in its mass matrix being large, so thegauge symmetry is broken;b. since the unphysical d.o.f (i.e.,tachyon/instanton/phantom) σ in (81) was excitednow, the vacuum tunnelling (oscillating) effect would become strong, so the off-diagonalelements in the mass matrix of A ′ i become large, too; or, in another viewpoint, now it’s A ′ µ that was frozen, and the tachyon was the real d.o.f for mediating interactions; we cantreat the tachyon massless or nearly massless according to the absence of heavy bosonsin a hadron;c. the VEV h A i i in the neighbour minimum are nearly equal, so, there would be adegenerate for the fermion mass, or, we can say, the flavor symmetry for fermions wouldbe restored; besides, it’s now allowed for very small fermion masses through (88), whichmight be an underlying reason for the feasibility of the “large N c ” or “large N f ” hypothesisfor a real hadron, and for the possible neutrino oscillation.So, maybe this is a new kind of dynamical symmetry breaking/restoring mechanism,with a seesaw for gauge symmetry and flavor symmetry. Instead of the gauge field A µ ∼ u∂u † ( u is a group element), the employment of theWilson line U ( y, x ) and Wilson loop U P ( x, x ), which are defined as [2] U P ( x + ǫn, x ) = 1 − igǫn µ A µ ( x ) + O (( gǫ ) ) , (91) U P ij ( x, x ) = 1 − iǫ gF ij + O ( ǫ ) , (92)ensured the availability of lattice gauge theory. It is just this subtle hint that inspired usto consider a field U , with a hidden correspondence of the Wilson loop U P , U P → U , (93)rather than the gauge field A as a possible effective d.o.f., with the Light Constraint in(85) gA µ = u ( x ) i∂ µ u † ( x ) → U ( x ) i∂ µ U † ( x ) , . (94)Thus, as an inverse procedure, it is a useful try to solve the non-perturbative problem instong QED by defining a P4 type complex scalar field U = U + iU with U and U areboth excited, instead of the group element u .
2. Media fields are P2 type, while matter fields are underlying P4 type.
Besides the media field A , we can also treat the fermion matter field ψ as P4 typefield. For convenience, we choose a scalar matter field φ and take the scalar QED as anexample to illustrate our motivation. 16f we treat the field φ as effective reduction of underlying P4 type field Φ, then the P2type current of φ will become a P2 type field, as J µ ( x ) = φ † i∂ µ φ ( x ) → Φ † i∂ µ Φ( x ) ≡ A µ ( x ) , (P2 type field φ → P4 type field Φ) → (Maurer-Cartan 1-from of Φ) , (P2 type current J µ ) → (P2 type field) . (95)It is reasonable for (95), since the only difference between a current and a vector field isthat: a field has a E.O.M, while a current hasn’t; for other things, they could be treatedas the same.Thus, with the Light Constraint in (85), the old P2 type (nonrenormalizable) 3-particleinteraction term will become a new 2-particle mixing term (which will be a perturbative“interaction term”), as L I = eA µ φi ←→ ∂ µ φ → eA µ Φ i ←→ ∂ µ Φ = eA µ A µ = L K . (96)Besides, it seems like that the new d.o.f. A µ in the limit of φ → Φ could propagateto a composite system of two collinear φ , so, the generation of the new mixing term (or,the new d.o.f. A µ ) is associate with collinear motion of the two φ particle, which is alsoa kind of “kinetic-potential duality”.
3. On the generalization of a theory
On the generalization of a theory, one method might be to extend the d.o.f, such as, tointroduce greater symmetry, more particles and more interactions, more extra dimensions,or more complicate rules, mathematically, by introducing more complicate groups, morecomplicate variables (e.g., complex, quaternion or octonion valued), more coordinates,higher-order and nonlinear equations, etc. If the results in our calculations are usefulfor the real physical processes, then it would be said that the P4 type theory is a moregeneral theory than the P2 type ones.Another method might be to redefine the effective d.o.f for a system, such as: theBogoliubov transformation; the wave-particle duality, in which the redefinition of thecanonic d.o.f from the momentum (current) p to the wave ( p → φ ) for the first quantizationin quantum mechanics; or, from the P4 type current to the P2 type field ( J = Φ i∂ Φ → A )in this paper. According to these examples, maybe we could ask, is there a principleabout this redefinition of d.o.f, or maybe we can call it “materialization” (from variableto matter)?Besides, when the dynamics of a variable (or, d.o.f) becomes complicate, maybe it isthe time to redefine the kinetics rather than the interactions for the systems, such as thechaos or the turbulence systems, or non-perturbative or the nonrenormalizable systems.Or, maybe we can give a hypothesis that, all the good (or, well-defined) theories should besimple (or, perturbative, linear), while all the complicate (or, non-perturbative, nonlinear)theories should be due to the bad choice of the d.o.f.17 The causality in our P4 theory
Firstly, let’s solve the E.O.M in (29). For simplicity, we will only consider the U compo-nent of U = U + iU in the case of Λ U = 0, i.e., − ∂ µ ∂ ν ∂ µ ∂ ν U = − m U U , p = m U . (97)Then, by combining the plane-wave solution (21) and the instanton solution (24), wemight get the general solution with the form of U ( x ) = c e ip · x + c e − ip · x + d e p · x + d e − p · x , (98)however, we will write the general solution with the form of U ( x ) = αa p e − ip · x + α † a † p e ip · x , (99)that is to say, the effects of the unphysical solution are absorbed into the coefficients α and α † .Secondly, we will introduce a new postulation for the canonic quantization of our P4type field U . By taking the U component of U = U + iU as an example, we can expressthe canonic quantization results as below:(1) canonic variables of field U (with Eq. (99)): U ( x ) ≡ Z d p (2 π ) p p p (cid:0) α p a p e − ip · x + α † p a † p e ip · x (cid:1) , (100)Π ( x ) ≡ ∂ t ∂ U ( x ) = Z d p (2 π ) i ( p · p ) p p (cid:0) α p a p e − ip · x − α † p a † p e ip · x (cid:1) ; (101)(2) the creation and annihilation operators, state vectors and normalization relations:¯ p ≡ ( p ) ± m , with p ≡ p · p = ± m ; (102) α | i ≡ , | ∗ i ≡ s ( p · p ) ¯ p ( p ) α † p | i , h | i = h ∗ | ∗ i = 1; a | i ≡ , | p i ≡ p p a † p | i , | p ∗ i ≡ p p a † p | ∗ i , h p | q i = h p ∗ | q ∗ i = 2 p (2 π ) δ (3) ( p − q ); (103) h | U ( x ) | p ∗ i = 1 p e − ip · x ; (104)that means, we define the operator α p or α † p with the operation of shifting a stable physicalvacuum | i to another unstable unphysical vacuum | ∗ i , and the effect of field operator U is to annihilate an unstable state | p ∗ i created from an arbitrary unstable vacuum | ∗ i ;and, in the limit case that α p and α † p are c -numbers, only the physical vacuum | i andthe physical states | p i exist; 183) canonic commutators:[ U ( x , t ) , Π ( x ′ , t )] = (cid:2) U ( x , t ) , ∂ t ∂ U ( x ′ , t ) (cid:3) = − iδ (3) ( x − x ′ ) · ( p ) ¯ p , ⇔ h α p , α † p ′ i = ( p ) ( p · p ) ¯ p , h a p , a † p ′ i = (2 π ) δ (3) ( p − p ′ ) , h α p a p , α † p ′ a † p ′ i = (2 π ) δ (3) ( p − p ′ ) ( p ) ( p · p ) ¯ p ,others = 0; (105)(4) Legendre transform and Hamiltonian:as the Lagrangian can be rewritten to be L U = − ∂ µ ∂ ν U ( x , t ) ∂ µ ∂ ν U ( x , t ) + m U [ U ( x , t )] = ∂ ν U ( x , t ) ∂ ν [ ∂ U ( x , t )] − ∂ µ [ ∂ ν U ( x , t ) ∂ µ ∂ ν U ( x , t )] + m U [ U ( x , t )] = ˙ U ( x , t ) ∂ t [ ∂ U ( x , t )] − ∇ U ( x , t ) · ∇ [ ∂ U ( x , t )] + m U [ U ( x , t )] , (106)where the second term (the total derivative) in the second line in (106) can be droppedbecause the corresponding surface term is zero on the boundary of the space, the Hamil-tonian can be get from the Legendre transform H U = − Z d x h Π ( x , t ) ˙ U ( x , t ) − L U ( x , t ) i = + Z d p (2 π ) (cid:20) ( p · p ) ¯ p ( p ) (cid:21) (cid:0) α † p a † p α p a p + α p a p α † p a † p (cid:1) = + Z d p (2 π ) p (cid:0) a † p a p + a p a † p (cid:1) , (107)and the Schrodinger equation can be get as H U | p i = p | p i . (108)Thirdly, we want to emphasize that, although there are fourth-order derivative termsand unphysical solutions in our P4 type field theory, however, (1) according to the Hamil-tonian in (107), H ∼ p = ( ˙ p ) = ∂ t x , the corresponding classic dynamics will in-clude only 2nd-order derivative terms (with p p ∼ p as “cononical momentum” and p ∼ ( p ) / p in the Hamiltonian to get an E.O.M of the form p ¨ x + κ x = 0) so thatthere will not be acausality; (2) only two canonical variables, U and ∂ t ∂ U , are enoughto construct our theory, that is to say, it is not necessary to define multiple canonicvariables, e.g., extra extended conjugate momenta such as ˙ U and ¨ U , although they areneeded as initial conditions to fix the 4 coefficients in (98) or (99) in the classic mechanics. Let’s go back to the causality topic mentioned in Section 2.3. Here are the different ex-pressions to causality in classic mechanics and quantum mechanics:(a) in classic mechanics, the causality depends on the interval of the variables in thecoordinate space, i.e., whether the interval is time-like or not;19b) in the
Heisenberg picture for quantum mechanics, the causality depends on the“interval” (defined by the commutator) of the variables (operators) in the algebra space,i.e., whether the “interval” is time-like or not;(c) in the path integral formalism, the causality depends on the time-order operatorˆT inserted for the Feynman propagators due to the retard potential boundary condition;etc.We want to propose that unphysical d.o.f does not mean acausality. In our P4 typefield theory formalism, the causality is rigid, since the correlation function defined in (31)is also expressed with the time-order operator ˆT! What we need to do is only to interpretthe effects of the unphysical d.o.f.Here we rewrite the correlation function in (31) as D F ( x − y ) ≡ h Ω | ˆT U ( x ) U ( y ) | Ω i , (109)where | Ω i and | Ω i are two vacuum states. If we write the “interval” between two vacuumstates | Ω i and | Ω i as their inner product h Ω | Ω i ≡ e iθ , the value of θ can be real orcomplex. A real-valued θ is associated with the vacuum tunnelling processes among theso-called stable “ θ vacuum”, and the complex-valued θ would be associated with the moregeneral vacuum evolution dynamics. Our postulation is that, effects from the unphysicald.o.f of U ( x ) and the non-unitary vacuum transition processes are combined to a physicalpropagator.The unphysical things are indeed physical. Some examples are listed as below:(1) In the “unphysical limit”, i.e., V ( U ) = − m U U † U = + λ U Λ U U † U U † U = 0 in(25), the fields U and U (or, σ ( x ) and φ ( x )) in (81) are both excited, and now, thehiggs/tachyon/instanton/phantom effects are excited completely, which will be reflectedin the detectable world. Now we might understand the global U (1) symmetry of U fieldas a kind of symmetry between the inner region and the outer region of the light cone.(2) In the “physical limit”, or the“Light constraint”, U + U = h U i , see (85) inSect. 5.3.1, that is, in the vacuum symmetry breaking case, the gauge symmetry wouldarise automatically; the vacuum states are all stable, and, only the physical d.o.f e − ip · x and speed value c = 1 for the light survive as t → ∞ , so their effects can be physical anddetectable all the time.(3) In the “meta-physical case”, if the “Light constraint” is not rigidly satisfied, thenthe unphysical partner of light would exist and the speed of light would fluctuate; althoughthe so-called unphysical d.o.f e − p · x in (22,23) are unstable, their residual effect should bedetectable until t → ∞ (no matter the momentum p of field U is large or small). Inother words, nontrivial vacuum could be treated as potential barrier background, so anattenuation (imaginary-valued momentum) is normal for a particle transit through thepotential barrier.By combining the interpretation to causality above, maybe we can introduce a newterminology called “ propagator picture ”, that is, we interpret the causality by intro-ducing a generalized path integral formalism, including the unphysical particle d.o.f andthe unphysical vacuum but generating physically causal amplitudes. In a word, the inclu-sion of the unphysical particle d.o.f and vacuum evolution dynamics is the origin of thedifferences between our P4 type field theory and the P2 type theories.20 Conclusion
We have introduced a new class of higgs type complex-valued scalar fields U (“P4 type”)with a fourth-order differential equation as its equation of motion, motivated by the linearpotential in the lattice gauge theory, and we have seen something new in a theory whichcan generate a linear potential on the level of effective theories. The field U can generate awealth of interaction forms with some postulations on the convergence being taken. Aftergetting a propagator of the form of − i/p from a ( ∂∂U ) term in the kinetics term inthe canonic quantization formalism, by computing the amplitudes of the tree-level 2 → U field, we can get a wealth of classic non-relativisticeffective potential form within the Born-approximation formalism, such as: (1) by using U to construct a QED theory, we can get the Coulomb-type potential, with a negligiblelinear potential and logarithmic potential as correction; (2) by using U to construct a QCDtheory, we can get the Coulomb-type potential, and a considerable linear potential to servefor the confinement, with a logarithmic potential as the next-leading order corrections;(3) by using U to construct a gravatition theory, we can get a linear potential to servefor the dark energy effect and the inflation effect, a Coulomb-like potential to serve forthe Newton’s gravity, and a logarithmic potential combining with a relativistic correctionto Newton’s gravity to serve for the dark matter effect; in the sense of the superficialdegree of divergence, this gravitation theory is renormalizable, so, a construction of arenormalizable gravitation theory might be practicable in our P4 type formalism.Moreover, in some limit cases, we can get some interesting deductions, such as: (1)in a low energy approximation of the dynamics of U , a nonlinear Klein-Gordon equationcould be generated; (2) with a constraint U + U = h U i to a spontaneous breaking U (1)symmetry, U could become a group element, thus the gauge symmetry could superficiallyarise, with a linear QED to be generated by relating the field strength ∂U to the corre-sponding gauge field A µ ; (3) due to the multi-vacuum structure for a sine-Gordon typevector field A µ induced from U , a mass spectrum with generation structure and a seesawmechanism on gauge symmetry and flavor symmetry could be generated, including heavyparticles; (4) by treating the P2 type matter fields as the effective d.o.f of P4 type ones(with a kind of “kinetic-potential duality”), or, by treating the P2 type gauge field A µ asthe effective d.o.f of P4 type U fields (with a correspondence to the Wilson line U P ), itprovides a possible way to deal with the non-perturbative problems. So, a solution to thenon-perturbative problems might be practicable in our P4 type formalism.For the causality, we interpret the causality by introducing a generalized path integralformalism, including the unphysical particle d.o.f and the unphysical vacuum but gener-ating physically causal amplitudes. In a word, the inclusion of the unphysical particled.o.f and vacuum evolution dynamics is the origin of the differences between our P4 typefield theory and the P2 type theories. The author is very grateful to Prof. Xin-Heng GUO at Beijing Normal University, Dr.Xing-Hua WU at Yulin Normal University and Dr. Jia-Jun Wu at University of ChineseAcademy of Sciences, for very important helps.21 eferences [1] C.Y. Wong, “
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