A quantum mechanism underlying the gauge symmetry in quantum electrodynamics
aa r X i v : . [ phy s i c s . g e n - ph ] J un A quantum mechanism underlying the gauge symmetry in quantum electrodynamics
Wen-ge Wang ∗ Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China (Dated: June 16, 2020)In this paper, a formulation, which is completely established on a quantum ground, is presentedfor basic contents of quantum electrodynamics (QED). This is done by moving away, from thefundamental level, the assumption that the spin space of bare photons should (effectively) possessthe same properties as those of free photons observed experimentally. Within this formulation,bare photons with zero momentum can not be neglected when constructing the photon field; andan explicit expression for the related part of the photon field is derived. When a local gaugetransformation is performed on the electron field, this expression predicts a change that turnsout to be equal to what the gauge symmetry requires for the gauge field. This gives an explicitmechanism, by which the photon field may change under gauge transformations in QED.
I. INTRODUCTION
Quantum field theory (QFT) supplies a framework forestablishing the most successful model for elementaryparticles, i.e., the standard model (SM) [1–3]. However,there are still some important problems unsolved withinQFT. In this paper, we discuss some aspects of two ofthem. One problem is that the physical origin of gaugesymmetry is still unclear. As is well known, to a large ex-tent, modern physics is established on two symmetries,the Lorentz symmetry and the gauge symmetry. Thephysical origin of the Lorentz symmetry is quite clear,i.e., the physics should not depend on the frame of refer-ence chosen. In contrast, presently, no physical mecha-nism is known for the gauge symmetries employed in theSM.Another problem is that, although elementary particlesare quantum particles in their nature, in the establish-ment of the SM, classical fields must be made use of at thefundamental level. Specifically, in the canonical quanti-zation formalism, quantum fields are introduced by quan-tizing classical fields; and in the path-integral quantiza-tion formalism integrals are computed for all realizationsof classical fields. Efforts have been seen in the directionof putting QFT completely on a quantum ground (see,e.g., Ref.[1]), however, serious difficulties still lie in theway. In this paper, we go further in this direction, bysimplifying some of the strategies and methods used inRef.[1]. Interestingly, with some progresses achieved inthis direction, a clue is seen for a partial solution to thefirst problem discussed above.Basically, Ref.[1] suggests a route consisting of threesteps: (i) establishing the quantum state space, (ii) con-structing the quantum fields, and (iii) building the totalLagrangian. A method of accomplishing the first step isdiscussed there, however, hard difficulties are met at thesecond step with quantum gauge fields. In order to over-come these difficulties, help is asked for classical fieldsin a way similar to the ordinary treatments; that is, the ∗ Email address: [email protected] quantum gauge fields are obtained by quantizing classicalfields with certain gauge fixing.Examining the hard difficulties mentioned above, weobserve that they are related to an assumption made atthe fundamental level of the theory, which states thatthe spin spaces of bare bosons should (effectively) pos-sess the same properties as those of free bosons observedexperimentally (see Sec.II for detailed discussions). Sincein fact there is no generic physical principle that may liebehind this assumption, it is of interest to study what thetheory may look like, if this assumption is moved awayfrom the fundamental level of the theory. To study thispossibility, in this paper, we consider the simplest partof the SM, namely, the quantum electrodynamics (QED).We are to show that no serious difficulty is met with thethree-step route suggested in Ref.[1], if this assumptionis moved away.Moreover, within the framework of QED thus estab-lished, a clue is seen for a deeper understanding of gaugesymmetry. In fact, within it, the photon field constructedin the usual way is not complete — lacking the contribu-tion from bare photons with zero momentum. We foundthat the only reasonable way of constructing the part ofthe field for these bare photons is to make use of thevacuum fluctuations induced by them. Going further, anexplicit expression of this part of the field is derived. Itpredicts that, under a local U (1) gauge transformationperformed on the electron field, this part of the photonfield should undergo a change, which is equal to what isrequired by the gauge symmetry for the transformationof the total gauge field.The paper is organized as follows. In Sec.II, we an-alyze some difficulties met in the route of establishingQFT discussed in Ref. [1], then, propose a strategy ofmodifying it. In Secs.III, following the proposed strat-egy, we formulate the basic part of QED. In Sec.IV, wederive an explicit expression for the part of the photonfield that is related to bare photons with zero momen-tum and, then, show that it supplies a mechanism forunderstanding gauge transformations of the gauge field.Finally, conclusions and discussions are given in Sec.V. II. PRELIMINARY ANALYSIS
In this section, we analyze some difficulties met withthe above-discussed three-step route suggested in Ref.[1]in the attempt of establishing QFT on a quantum ground.We also discuss some method for improvement.We begin with recalling some details of the first twosteps given in Ref.[1]. At the first step, the state space isestablished. It is spanned, as usual, by states that are ob-tained from all possible direct products of single-particlestates of all particle species. Here, a single-particle statepossesses a momentum part and a spin part; the mo-mentum part is just given by an ordinary momentumstate, while, properties of the spin part may be species-dependent. In Ref. [1], properties of the spin part are de-termined by a mathematical method of induced represen-tations [4], together with the well-known spin-statisticsrelationship [5, 6] and an assumption mentioned in Sec.I. In what follows, we indicate this assumption by A bbs ,with “bbs” standing for “bare boson spin”. • A bbs . The spin spaces of bare bosons should pos-sess the same properties as those of the related freebosons observed experimentally, at least effectivelyso.At the second step, quantum fields are constructed.To do this, creation and annihilation operators are intro-duced from the particles states gotten at the first step.The quantum fields are written in the ordinary way asintegrals of products of creation and annihilation oper-ators and single-particle wave functions. For fermions,the quantum fields can be directly constructed, withoutmeeting any serious problem.However, hard difficulties are met with bosons subjectto the assumption A bbs . In fact, on one hand, polariza-tion vectors have four components, on the other hand,the angular momentum of experimentally-observed W ± -boson (same for Z -boson) has only three independenteigenstates. Even worse, no four-vector quantum fieldcan be constructed for photons, if they are required topossess only two independent polarization states. Thesedifficulties are so hard that, in order to circumvent them,Ref. [1] goes back to the ordinary treatments to QFT, inwhich quantum fields are obtained by quantizing classicalfields.Now, we give analysis. From the above discussions, it isseen that the emergence of the above-discussed hard dif-ficulties is related to the assumption A bbs . One may notethat there is, in fact, no generic physical principle thatthis assumption may be based on. On the contrary, fromthe conceptual angle, the assumed relationship looks un-usual, because the concept of bare particle lies at the As an interesting feature of this treatment, the Dirac equationfor fermionic spin states emerges as a requirement from symme-try considerations. fundamental level of the theory, while, this is unnecessaryfor the concept of experimentally-observed free particle.One may see the above-mentioned point about con-cept clearly in an imagined model, which contains nodivergence. In such a model, properties of “bare parti-cles” are assumed at the fundamental level. Their statesgive a basis, in which matrices of Hamiltonians are writ-ten. Meanwhile, stable states of ”free particles” observedexperimentally are usually interpreted as eigenstates ofsome Hamiltonians.In a QFT such as the SM, the situation is much morecomplicated than the imagined model, due to various di-vergences, particularly the ultraviolet divergence. Forthe purpose of giving finite predictions for experimentalresults, renormalization is needed to move out the di-vergences. As is known, the assumption A bbs is indeeduseful in the computation of the scattering matrix, whoseelements are for states of free particles observed experi-mentally. In fact, in order to get finite predictions in amodel suffering of the ultraviolet divergence, it is alwaysnecessary to introduce some assumption like A bbs , as aworking assumption, which imposes a link between spinspaces at the fundamental level of the theory and thoseof free particles observed experimentally.A relevant question is about the stage, at which suchan assumption may be introduced. One method is to doit at the fundamental level, like in ordinary formulationsof QFT. This may give the theory a relatively good ap-pearance, but, usually at the cost of imposing a quitecomplicated and involved structure at the fundamentallevel, even at a risk of blurring some actual structure ofthe theory.In order to avoid the above-mentioned flaw, one mayconsider an alternative strategy, in which such an as-sumption is put at some level higher than the fundamen-tal level. In other words, such an assumption may beintroduced, after the fundamental part of the theory is es-tablished. Partially, this strategy is similar that adoptedin the indefinite-metric scheme of canonical quantizationof the classical electromagnetic field [8]. Summarizing the above discussions, we propose to con-sider the following route, as a modification to the three-step route discussed in Ref.[1].R1. At the fundamental level.(a) Establishing the quantum state space,(b) constructing the quantum fields,(c) building the total Lagrangian.R2. At a higher level. • Either introducing an assumption like A bbs , For a difference between our treatment and the indefinite-metricscheme, see discussions given in the paragraph below Eq.(17). • or interpreting eigenstates of some effectiveHamiltonians, if obtainable, as states of freeparticles observed experimentally.In this paper, we discuss only at the fundamental leveland, hence, we do not need to consider any assumptionlike A bbs . III. BASIC CONTENTS OF QEDOM
In this section, we discuss basic contents of QED ata fundamental level, not including the assumption A bbs .We first give some preliminary discussions in Sec.III A,then, discuss the quantum state spaces and quantumfields in Sec.III B and Sec.III C, from which the ordinaryLagrangian can be built (Sec.III D). A. Preliminary discussions
We first give some words for terminology. Since bareelementary particles are quite different, in their nature,from free particles that are observed experimentally, itwould be convenient to give them names with more ex-plicit distinction. For this reason, we use the term mode to refer to a bare elementary particle that is consideredat the fundamental level of QED; meanwhile, free parti-cles observed experimentally are just called free particles.(A further reason of making this distinction will be dis-cussed in Sec.V.) Specifically, instead of bare electron,bare positron, and bare photon, we say e -mode, e -mode(called anti e -mode), and A -mode. Moreover, with theassumption A bbs moved away from the fundamental level,the formulation of the fundamental part of QED to begiven below is not exactly the same as ordinary ones. Weuse “ QED of modes ”, in short, QEDoM, to refer to theformulation of QED to be given below.Like in the ordinary treatment, the state of an arbi-trary mode is assumed to possess two parts, a momen-tum part and a spin part. We use E M to denote the statespace of a single mode M , with M = e, e, A , and, asusual, assume that it has the following form, E M = M p | p i ⊗ S M , (1)where | p i indicate the ordinary momentum states and S M is either a representation space of the Lorentz groupor a subspace of it. The momentum states satisfy anordinary normalization condition, i.e., h q | p i = p δ ( p − q ) , (2)where p = p | p | + m , with m indicating the mass of e -mode. The spin space S M may be a function of themomentum p .In Ref. [1], to satisfy the assumption A bbs , properties ofthe spaces S M are determined by the method of induced representations, which makes use of representations ofthe so-called little group [4]. But, in QEDoM, with theassumption A bbs moved away from the fundamental level,there is no need of using the method of little group to de-termine properties of S M . In fact, the physical reason ofemploying the method of little group is far from beingapparent. Instead, to determine properties of S M , weare to employ an alternative method that was discussedin Ref.[7]; this method is based on the physical require-ment that predictions for experimental results must havedefinite and real values. B. States and fields of e -mode and e -mode In this section, we first discuss the state space E e fora single e -mode, then, discuss E e for an e -mode. Finally,we construct the corresponding quantum fields.We assume that the spin space S e should satisfy two re-quirements: (i) being a subspace of the four-dimensionalDirac-spinor space and (ii) being a Hilbert space. As iswell known, a Hilbert space is a complete and complexlinear space equipped with an inner product. For two ar-bitrary vectors | ψ i and | φ i , their inner product, denotedby ( | ψ i , | φ i ), by definition should possess the followingbasic properties,( | ψ i , a | φ i + b | φ i ) = a ( | ψ i , | φ i ) + b ( | ψ i , | φ i ) , (3a)( | ψ i , | φ i ) = ( | φ i , | ψ i ) ∗ , (3b)( | ψ i , | ψ i ) ≥ | ψ i , | ψ i ) = 0 iff | ψ i = 0 , (3c)where a and b are arbitrary complex numbers. A physicalidea underlying Eqs.(3b)-(3c) is to ensure that the theorymay give predictions that take real and definite values.Spaces that satisfy the two requirements mentionedabove have been discussed in Ref.[7] for a fermion with anonzero mass. According to results given there, the sim-plest and most natural choice of S e is that it is spanned bythe two ordinarily-used Dirac spinors U r ( p ) of r = 0 , These U -spinors satisfy the following normalizationcondition, U † r ( p ) γ U s ( p ) = 2 m δ rs . (4)Thus, a basis of E e , denoted by | e p r i , is written as | e p r i = U r ( p ) | p i . (5)The label r is raised by δ rs and lowered by δ rs . The braof | e p r i is defined in the ordinary way, that is, h e p r | = In this treatment, the Dirac equation for spin states appears asthe simplest and most natural condition, under which a subspaceof the four-dimensional Dirac-spinor space is a Hilbert space. More consistently, one may write the spinors U r ( p ) in the ab-stract notation of Dirac’s ket as discussed in Ref. [7], such that | e p r i = | U r ( p ) i| p i . h p | U † r ( p ) γ . As a result, h e p r | e q s i = 2 m p δ ( p − q ) δ rs . (6)Next, the e -mode can be treated in a similar way. Thespace S e ( p ) is spanned by the two ordinarily-used Diracspinors V r ( p ) of r = 0 ,
1, which satisfy the following nor-malization condition, V † r ( p ) γ V s ( p ) = − m δ rs . (7)Due to the minus sign on the right-hand side (rhs) ofEq.(7), the space S e ( p ) is not exactly a Hilbert space.But, since this minus sign appears for both of the twospinors V r ( p ), practically, its effects can be easily movedaway, without affecting the physical idea of obtaining fi-nite and real predictions. Then, the state of a single e -mode with a momentum p is written as | e p r i = V r ( p ) | p i , (8)and the corresponding bra is written as h e p r | = h p | V † r ( p ) γ .Making use of direct products of the single-mode statesdiscussed above, it is easy to construct the total statespaces for e -mode and e -mode. Then, creation and anni-hilation operators for these two modes can be introducedby an ordinary method [1]. Obeying the spin-statisticsrelationship, they satisfy the following well-known anti-commutation relations, { b r † ( p ) , b s † ( q ) } = 0 , (9a) { d r † ( p ) , d s † ( q ) } = 0 , (9b) { b r † ( p ) , d s † ( q ) } = 0 , (9c) { b r ( p ) , b s † ( q ) } = p δ rs δ ( p − q ) , (9d) { d r ( p ) , d s † ( q ) } = p δ rs δ ( p − q ) . (9e)Note that there is no minus sign on the rhs of Eq.(9e).For single-mode states, one writes | e p r i = b r † ( p ) | i , h e p r | = h | b r ( p ) , (10a) | e p r i = d r † ( p ) | i , h e p r | = h | d r ( p ) , (10b)where | i indicates the vacuum state.Finally, we construct quantum fields for e -mode and e -mode. A standard construction is given by [1] ψ ( x ) = Z d e p (cid:0) b r ( p ) U r ( p ) e − ipx + d r † ( p ) V r ( p ) e ipx (cid:1) , (11a) ψ † ( x ) = Z d e p (cid:0) b r † ( p ) U † r ( p ) e ipx + d r ( p ) V † r ( p ) e − ipx (cid:1) , (11b)where d e p = p d p . Here and hereafter, by convention,double appearance of a same index, one in an upper posi-tion and the other in a lower position, implies summation In the literature, the factor p in d e p is sometimes written as over the index. Like in the standard treatment, oneassumes that the Lagrangian density for free e -mode andfree e -mode, denoted by L ee , has the form of L ee = ψ † γ ( iγ µ ∂ µ − m ) ψ, (12)where ∂ µ ≡ ∂/∂x µ . C. States and fields of A -mode In this section, we discuss the state space E A for asingle A -mode, then, discuss the corresponding quantumfield.The spin space S A is assumed to be a four-componentvector space, denoted by V , which possesses a metric g µν with diagonal elements [ g µµ ] = [1 , − , − , −
1] and offdi-agonal elements zero. Clearly, it is not a Hilbert space.An often-used basis of V is given by the polarization vec-tors ε µλ ( k ), corresponding to a given momentum k ; theysatisfy ε ∗ λ,µ ( k ) ε µλ ′ ( k ) = g λλ ′ . Then, the state space E A for one A -mode is spanned by the following states, | A k λ i = ε µλ ( k ) | k i . (13)The bra of | A k λ i is defined in the ordinary way, that is, h A k λ | = h k | ε ∗ λµ ( k ) . (14)This gives that h A k λ | A k ′ λ ′ i = g λλ ′ k δ ( k − k ′ ) , (15)where k = | k | .Making use of the above-discussed states of single A -mode, one may easily construct the total state space for A -modes. One may also introduce creation and annihila-tion operators, denoted by a † λ ( k ) and a λ ( k ), respectively,which satisfy the following commutation relations,[ a † λ ( k ) , a † λ ′ ( k ′ )] = 0 , (16a)[ a λ ( k ) , a † λ ′ ( k ′ )] = g λλ ′ k δ ( k − k ′ ) . (16b)Without the need of obeying the assumption A bbs , thefield for the A -mode, denoted by A µ ( x ), can be directlyconstructed, A µ ( x ) = Z d e ka λ ( k ) ε λµ ( k ) e − ikx + a † λ ( k ) ε λ ∗ µ ( k ) e ikx , (17) √ p . Here, we write this form of d e p , because it is Lorentz-invariant. Consistently, the anti-commutation relation for cre-ation and annihilation operators has a factor p [see Eq.(9)].Some constant prefactor may be multiplied to the field ψ , whichis not written explicitly for brevity. In this paper, repeated labels in a same type of position do notimply summation. where d e k = k d k .One meets a normalization problem in the state space E A ; that is, states in it can not be normalized in the or-dinary way by making use of the scalar products given inEq.(15), due to the inhomogeneous sign of g λλ . In fact,a similar problem is met in the indefinite-metric schemeof canonical quantization of the classical electromagneticfield [8]; there, the problem is solved by imposing anauxiliary condition related to a specific gauge, which ef-fectively imposes a restriction to the physical state space.But, this method is not applicable here, because at thefundamental level of the theory we intend to formulateQEDoM in a gauge-independent way.We solve the above-discussed problem by a method dis-cussed in Ref.[7]. As mentioned previously, the physicalidea, which underlies Eqs.(3b)-(3c) for an inner prod-uct in a Hilbert space, is to give real and definite valuesfor predictions (for measurement results). As discussedin Ref.[7], for the purpose of achieving this goal, thesetwo equations are in fact too restrictive and one may, in-stead, consider a generalized inner product . For two arbi-trary vectors | ψ i and | φ i , their generalized inner product( | ψ i , | φ i ) possesses the following properties,( | ψ i , a | φ i + b | φ i ) = a ( | ψ i , | φ i ) + b ( | ψ i , | φ i ) , (18a)( | ψ i , | φ i ) ∗ = β ( | φ i , | ψ i ) , (18b)( | ψ i , P| ψ i ) ≥ | ψ i , P| ψ i ) = 0 iff | ψ i = 0 , (18c)where β is a parameter and P is a Lorentz-invariant op-erator. Clearly, at β = 1 and P = I being the identityoperator, the generalized inner product reduces to theordinary inner product.In an explicit construction of a generalized inner prod-uct for the A -mode, we consider the simplest choice of β ,i.e., β = 1. Let us focus on the space E A , because gener-alization of the discussions to be given below to the totalstate space for A -mode is straightforward. A generic vec-tor in the space E A is written as | ψ i = R d e kc λ k | A k λ i , with c -number coefficients c λ k . The corresponding bra is givenby h ψ | = Z d e k h A k λ | c λ ∗ k . (19)We define the symbol ( | ψ i , | φ i ) in Eq.(18) in the ordinaryway, that is, ( | ψ i , | φ i ) = h ψ | φ i . (20)An operators P that satisfies Eq.(18c) may be con-structed by a method used in Ref.[7], that is, P = X λ Z d e k | A k λ ih A k λ | . (21)Since both d e k and the label λ are Lorentz invariant, thisoperator P is Lorentz invariant. Making use of Eq.(15),it is straightforward to verify that h A k λ |P| A k λ i = δ λ λ k δ ( k − k ) . (22) Then, it is easy to verify that Eq.(18c) is satisfied withthe operator P in Eq.(21) and, hence, the space E A pos-sesses a generalized inner product.The generalized inner product defined with the aboveoperator P is Lorentz invariant. To see this point, letus consider another arbitrary vector | φ i , expanded as | φ i = R d e kC λ k | A k λ i . Making use of Eqs.(20)-(22), it isstraightforward to find that( | ψ i , P| φ i ) = X λ Z d e kc λ ∗ k C λ k . (23)One notes that, under a Lorentz transformation, the mo-mentum label k of a coefficient, say, of c λ k , changes, while,the value of c λ k does not change. Then, from the rhs ofEq.(23), one sees that the value of ( | ψ i , P| φ i ) should beLorentz invariant. Based on the above discussions, thevector | ψ i may be normalized in the following way, X λ Z d e k | c λ k | = 1 . (24) D. The interaction Lagrangian for ψ ( x ) and A µ ( x ) The interaction between the fields constructed in theprevious sections can be introduced in the standard way,as briefly discussed below, by assuming the invariance ofthe Lagrangian density under local U (1) gauge transfor-mations of the e - e -mode fields. We use tilde to indicateresults of gauge transformations.Local U (1) gauge transformations of the e - e -modefields take the following form, ψ ( x ) → e ψ ( x ) = e − iθ ( x ) ψ ( x ) , (25a) ψ † ( x ) → e ψ † ( x ) = e iθ ( x ) ψ † ( x ) . (25b)Clearly, the Lagrangian density L ee for the e - e -mode fieldgiven in Eq.(12) is not invariant under the above trans-formations. To keep the total Lagrangian invariant, as iswell known, the A -mode field should undergo the follow-ing transformation, A µ ( x ) → e A µ ( x ) = A µ ( x ) − e ∂ µ θ ( x ) . (26)The total Lagrangian density denoted by L QED , whichhas the form of L QED = ψ † γ ( iγ µ D µ − m ) ψ −
14 ( F µν ) , (27)is invariant under the above gauge transformations.Here, D µ indicates the covariant derivative, D µ = ∂ µ − ieA µ ( x ) , (28)and F µν = ∂ µ A ν − ∂ ν A µ . (29)The interaction Lagrangian density is written as L int ( x ) = eψ † ( x ) γ γ µ ψ ( x ) A µ ( x ) . (30) IV. THE FIELD OF A -MODE WITH ZEROMOMENTUM In fact, the rhs of Eq.(17) is not complete in the con-struction of the quantum field for A -mode. This is be-cause the field should include all states that lie in thespace E A , while, the rhs of Eq.(17) does not include thestate of A -mode with zero momentum. In fact, the polar-ization vectors ε λµ ( k ) can not be defined for an A -modewith k µ = 0.In this section, we discuss the field of A -mode withzero momentum. For brevity, we call A -modes with zeromomentum null A -modes and use A np µ ( x ) to indicate thefield for null A -modes. Specifically, in Sec.IV A, we dis-cuss some properties that the null- A -mode field shouldpossess, then, in Sec.IV B, we discuss explicit expressionsfor the null- A -mode field. Finally, in Sec.IV C, we discusschanges of this field under local U (1) gauge transforma-tions of the e - e -field. A. Generic properties of the null- A -mode field It is natural to assume that the Lagrangian density forthe interaction between the null- A -mode field and the e - e -mode field, denoted by L npint ( x ), has the same form asthe ordinary one in Eq.(30), i.e., L npint ( x ) = eψ † ( x ) γ γ µ ψ ( x ) A np µ ( x ) . (31)This interaction Lagrangian implies that null A -modesmay generate quantum fluctuations as virtual pairs, eachconsisting of an e -mode and an e -mode.Since a null A -mode possesses zero energy, there is nophysical reason to assume that it may possess a nonzeroangular momentum. Hence, we assume that a null A -mode possesses no intrinsic degree of freedom. As aconsequence, a null A -mode has a one-dimensional statespace. We use a np and a np † to indicate the annihilationoperator and creation operator for null A -mode, respec-tively. Formally, one may write the null- A -mode fieldas A np µ ( x ) = N (cid:16) a np K µ ( x ) + a np † K ∗ µ ( x ) (cid:17) , (32)where N is a normalization factor and K µ ( x ) indicatessome c-number four-component vector field.Due to zero energy, there is in fact no restriction to thenumber of null A -modes that may exist. A reasonableassumption is that there may exist an infinite number ofnull A -modes. We use |∞ np i to indicate the normalizedstate of these null A -modes. The direct product of thisstate and the vacuum state | i is denoted by | np i , | np i ≡ | i ⊗ |∞ np i . (33)The infinite number of null A -modes requires that thetwo operators a np and a np † can not obey a commutationrelation like that in Eq.(16b). In fact, if one required that [ a np , a np † ] = 1, then, the vector a np |∞ np i would beunnormalizable.Since adding/subtracting one null A -mode to/from thestate |∞ np i should bring no change to it, the simplestassumption about the actions of a np and a np † is that a np |∞ np i = a np † |∞ np i = |∞ np i , (34)which is the only specific requirement that is imposedhere on the two operators a np and a np † . This assumptionrequires realness of K µ , i.e., K ∗ µ = K µ . Note that the twooperators a np and a np † should be commutable with allother creation and annihilation operators. Making use ofthis fact and Eqs.(32)-(34), one finds that A np µ ( x ) G | np i = 2 N K µ ( x ) G | np i , (35)where G represents an arbitrary function of operatorsthat do not include a np and a np † . Hence, the null- A -mode field A np µ ( x ) is effectively a c -number field and canbe written in the form of A np µ ( x ) = 2 N K µ ( x ). B. Explicit expressions of the null- A -mode field In order to find an explicit expression for K µ ( x ), onefaces the following obstacle: Null A -modes by them-selves possess no property that can be used to intro-duce a four-component vector. To solve this problem,the only conceivable method seems to make use of someeffect of the quantum fluctuations, which are induced bynull A -modes according to the interaction Lagrangianin Eq.(31). As discussed above, the fluctuations takethe form of emergence and vanishing of virtual e - e -modepairs.Based on discussions given above (in this and the pre-vious sections), we propose that the null- A -mode field A np µ ( x ) should possess the following properties.1. The c -number feature of the null- A -mode field im-plies that it may take the form of an expectationvalue of some operator in the state | np i .2. The above-mentioned operator describes emer-gence and vanishing of virtual e - e -mode pairs inquantum fluctuations and, hence, should containboth the e - e -mode field ψ ( x ) and its conjugate field ψ † ( x ).3. To construct a vector field from ψ ( x ) and ψ † ( x ),the simplest method is to make use of γ µ or ∂ µ .Since ψ ( x ) and ψ † ( x ) do not act on the null A -mode state |∞ np i , the null- A -mode field may in fact be written asan expectation value for the vacuum state | i .Then, there are two simplest and most natural candi-dates for expression of A np µ ( x ). The first one is A np µ ( x ) = N ( F (1) µ + F (2) µ ) , (36)where F (1) µ = h | ψ † ( x ) γ γ µ ψ ( x ) | i , (37a) F (2) µ = −h | Tr (cid:16) γ γ µ ψ ( x ) ψ † ( x ) (cid:17) | i ; (37b)and the second one is A np µ ( x ) = z N ( f (1) µ + f (2) µ ) , (38)where z is a parameter and f (1) µ = h | ψ † ( x ) γ (cid:0) ∂ µ ψ ( x ) (cid:1) | i , (39a) f (2) µ = −h | Tr (cid:16) γ ( ∂ µ ψ ( x )) ψ † ( x ) (cid:17) | i . (39b)As shown in Appendix A, under the plane-wave expan-sion of the e - e -mode field, the two expressions of A np µ ( x )in Eq.(36) and Eq.(38) are equivalent with z = i/m .Some remarks for F (1 , µ (similar for f (1 , µ ): (i) Theterm F (1) µ in fact describes an effect of emergence andvanishing of virtual e -mode, while, the term F (2) µ is forvirtual e -mode. (ii) The minus sign on the rhs of Eq.(37b)is due to the exchange of the order of ψ and ψ † , comparedwith that on the rhs of Eq.(37a).Substituting Eq.(11) into Eq.(39) and making use ofEq.(4), one finds that f (1) µ = i Z d e pV r † ( p ) γ p µ V r ( p ) = − im Z d e pp µ , (40a) f (2) µ = i Z d e qU r † ( q ) γ q µ U r ( q ) = 4 im Z d e qq µ . (40b)To deal with the integrals on the rhs of the two sube-quations of Eq.(40), one may employ a momentum regu-larization scheme. In this scheme, one considers a finitethree-momentum region with a cutoff Λ, i.e, with | p | < Λ[9], denoted by Ω(Λ). Under this regularization, f (1) µ and f (2) µ are written as f (1) µ, Λ = − im Z Ω(Λ) d e pp µ , (41a) f (2) µ, Λ = 4 im Z Ω(Λ) d e qq µ . (41b)Clearly, f (1) µ, Λ + f (2) µ, Λ = 0 and, as a result, one gets that f (1) µ + f (2) µ = 0 in the limit of Λ → ∞ . Hence, A np µ ( x ) = 0 . (42)(See Appendix B for a justification of Eq.(42) in view ofthe dynamics of virtual processes.) C. Null- A -mode field under gauge transformation As seen in Eq.(42), the null- A -mode field vanishes un-der the plane-wave expansion of the e - e -mode field given in Eq.(11). In this section, we show that this field getsfinite values under the gauge transformations of the e - e -mode field given in Eq.(25).With the null- A -mode field included, the total A -modefield, denoted by A tot µ ( x ), is written as A tot µ ( x ) = A µ ( x ) + A np µ ( x ) . (43)Correspondingly, the total Lagrangian density L totQED iswritten as L totQED = ψ † γ ( iγ µ D tot µ − m ) ψ −
14 ( F tot µν ) , (44)where D tot µ = ∂ µ − ieA tot µ ( x ) . (45) F tot µν = ∂ µ A tot ν − ∂ ν A tot µ . (46)In the previous section, two expressions of A np µ ( x ) weregiven [Eq.(36) and Eq.(38)], which are identical underthe plane-wave expansion of the e - e -field. It is easy tosee that these two expressions give different predictionsunder the local gauge transformation in Eq.(25). In fact,the rhs of Eq.(37) does not change under the transfor-mation and, hence, the expression in Eq.(36) predicts nochange of the null- A -mode field; in other words, it al-ways predicts a vanishing null- A -mode field. In contrast,Eq.(38) predicts that the field may get a finite value.To see which of the two expressions of A np µ ( x ) is appro-priate under the local gauge transformations in Eq.(25),let us consider a special case of the gauge transforma-tion, given by θ ( x ) = iqx with a constant vector q µ .Moving the gauge-phase terms of e − iqx and e iqx into theintegrations for the quantum fields of e ψ ( x ) and e ψ † ( x ),respectively, it is seen that the creation operator partfor the e -mode is written as b r † ( p ) U † r ( p ) e i ( p + q ) x , while,the corresponding part for the e -mode is written as d r † ( p ) V r ( p ) e i ( p − q ) x with a minus sign before the phaseterm iqx . Thus, this specific gauge transformation maybe interpreted as causing momentum shift for the e -modeand e -mode in the corresponding parts of their fields, andthe shift is different for these two modes.We recall that, in the plane-wave-expansion case dis-cussed in the previous section, the null- A -mode field van-ishes, meanwhile, the interaction Lagrangian L int ( x ) pre-dicts that the e -mode and e -mode generated from a null A -mode should possess opposite momenta. These twofacts, together with the above-discussed interpretationof that specific gauge transformation, suggests that thenull- A -mode should not definitely vanish under the gaugetransformation.Based on discussions given above, we assume thatEq.(38) is appropriate in the computation of the null- A -mode field under gauge transformations. SubstitutingEq.(25) into Eq.(39), straightforward derivation showsthat f (1) µ → e f (1) µ = f (1) µ + 4 m i ( ∂ µ θ ) Z d e p, (47a) f (2) µ → e f (2) µ = f (2) µ + 4 m i ( ∂ µ θ ) Z d e p. (47b)This gives that A np µ ( x ) → e A np µ ( x ) = − N (cid:0) ∂ µ θ ( x ) (cid:1) Z d e p. (48)Under the momentum regularization discussed previ-ously, the above transformation is written as A np µ, Λ ( x ) → e A np µ, Λ ( x ) = − N (cid:0) ∂ µ θ ( x ) (cid:1) Z Ω(Λ) d e p. (49)We set the normalization factor N as N = e Z Ω(Λ) d e p ! − . (50)Then, in the limit of Λ → ∞ , we get the following ex-pression of the transformed null- A -mode field, e A np µ ( x ) = − e ∂ µ θ ( x ) . (51)Equation (51) shows that the transformed null- A -modefield e A np µ ( x ) is equal to what is usually regarded as thegauge-symmetry-required change of the field A µ ( x ) [seeEq.(26)]. Since now A np µ ( x ) and A µ ( x ) form the totalfield A tot µ ( x ), in order to keep the total Lagrangian L totQED in Eq.(44) invariant under the gauge transformations ofthe e - e -mode field in Eq.(25), one needs to keep the field A µ ( x ) unchanged, that is, to assume that e A µ ( x ) = A µ ( x ) . (52)Thus, with the null- A -mode field A np µ ( x ) included, onegets a natural explanation to the gauge-symmetry re-quirement that the total gauge field should change bythe term − e ∂ µ θ ( x ), accompanying the gauge transfor-mation of the e - e -mode field in Eq.(25). In other words,the total Lagrangian written in the form of Eq.(44) natu-rally possesses the local U (1) gauge symmetry under thetransformation in Eq.(25). V. SUMMARY AND DISCUSSIONS
In this paper, we go further along a direction of linediscussed in Ref.[1], for the purpose of formulating QEDcompletely on a quantum ground. A key point of ourapproach is to move away an usually-adopted assump-tion from the fundamental level of the theory, which ba-sically states that the spin spaces of modes (bare par-ticles) should possess properties similar to those of the related free particles observed experimentally. With thisassumption released, a quite simple formulation of QEDis found at the fundamental level (without resorting toany classical field), which is done by three steps: (i) es-tablishing the quantum state space, (ii) constructing thequantum fields, and (iii) building the total Lagrangianby gauge symmetry.Within this formulation of QED, the photonic fieldincludes, in addition to the ordinarily-discussed part, anull- A -mode part that is related to A -modes (bare pho-tons) with zero momentum. An explicit expression ofthis part of the photonic field is derived, which reflectsa mean effect of quantum fluctuations in the vacuum. Itpredicts that, under local U (1) gauge transformations ofthe fermionic field, the null- A -mode part of the bosonicfield undergoes a change, which has the same form asthe well-known gauge-symmetry-required change of thegauge field. This suggests that the change of the pho-tonic field, which is required by gauge transformations,should come from its null- A -mode part. With this under-standing, the total Lagrangian naturally keeps invariantunder local U (1) gauge transformations of the fermionicfield.The above-discussed mechanism, by which the pho-tonic field changes under gauge transformations, is ne-glected in the ordinary formulations of QED. This isbecause there the photonic field is introduced by quan-tizing the classical electromagnetic field; while, as iswell known, the component of a classical electromagnetic(vector) field, which corresponds to zero frequency andzero wave length, has no physical significance. In this paper, we use the term “mode” to refer to whatis usually called bare elementary particle. One reasonof using this name was given previously, i.e., to makean explicit distinction from the concept of free particleobserved experimentally. Here, we give another reason,which is that the term mode may be assigned a moregeneric meaning.In fact, when quantum states of modes are introducedas the first step in the formulation of QED, they are la-belled with momentum and spin indexes; in other words,no definite feature in the configuration space is assignedto them. It is quantum fields, which are constructed atthe second step from wave functions of modes togetherwith creation-annihilation operators, that may be relatedto some specific feature(s) in the configuration space. Forexample, through a plane-wave expansion, one may getsome point feature. In principle, exploiting mode wavefunctions other than plane waves, it is possible to con-struct quantum fields with other features (say, a stringfeature). In fact, a hint may be seen even in the ordinary formulations ofQED. The hint is given by the fact that the bare-photon statewith zero momentum corresponds a singular point in the momen-tum space, since the integration over its neighborhood may givedivergent results for Feynman diagrams (infrared divergence).
In future investigation, it should be of interest tostudy whether the method used here may be general-ized to the electroweak theory with the gauge symmetry U (1) ⊗ SU (2). When doing this, one may consider a stageof the theory before the Higgs mechanism is used to in-troduce masses to vector bosons; at this stage, there aretwo species of massless neutral boson. Since the SU (2)gauge symmetry involves an intrinsic degree of freedom,such a generalization (if applicable) can not be a straight-forward one and some modification is expected. Acknowledgments
The author is grateful to Yan Gu and Hong Zhao forvaluable discussions and suggestions. This work was par-tially supported by the National Natural Science Founda-tion of China under Grant Nos. 11535011 and 11775210.
Appendix A: Relation between Eqs.(36) and (38)
In this appendix, we show that the two expressions inEqs.(36) and (38) are equivalent with z = i/m . Substi-tuting Eq.(11) into Eq.(37), one finds that F (1) µ = h | Z d e q (cid:0) b s † ( q ) U s † ( q ) e iqx + d s ( q ) V s † ( q ) e − iqx (cid:1) γ γ µ Z d e p (cid:0) b r ( p ) U r ( p ) e − ipx + d r † ( p ) V r ( p ) e ipx (cid:1) | i = h | Z d e pd e qd s ( q ) V s † ( q ) e − iqx γ γ µ d r † ( p ) V r ( p ) e ipx | i = Z d e pV r † ( p ) γ γ µ V r ( p ) . (A1)Similarly, F (2) µ = −h | Tr h γ γ µ Z d e p (cid:0) b r ( p ) U r ( p ) e − ipx + d r † ( p ) V r ( p ) × e ipx (cid:1) × Z d e q (cid:0) b s † ( q ) U s † ( q ) e iqx + d s ( q ) V s † ( q ) e − iqx (cid:1) i | i = − Z d e p Tr h γ γ µ U r ( p ) U r † ( p ) i = − Z d e p U r † ( p ) γ γ µ U r ( p ) (A2)The Dirac equation ( γ ν p ν + m ) V r ( p ) = 0 gives that V r ( p ) = − γ ν p ν V r ( p ) /m , (A3a) V r † ( p ) = − V r † γ ν † p ν /m . (A3b) Making use of Eq.(A3) and the relations that ( γ ) = 1and γ γ µ † γ = γ µ , one finds that V r † ( p ) γ γ µ V r ( p ) = − V r † γ γ µ γ ν p ν V r /m , (A4a) V r † ( p ) γ γ µ V r ( p ) = − V r † γ γ ν p ν γ µ V r /m . (A4b)Then, noting the relation { γ µ , γ ν } = 2 g µν and Eq.(4),one gets that V r † ( p ) γ γ µ V r ( p ) = − m V r † γ g µν p ν V r = 4 p µ . (A5)Following the same procedure as that given above,for the spinors U r ( p ), which satisfy the Dirac equation( γ µ p µ − m ) U r ( p ) = 0, one finds that U r † ( p ) γ γ µ U r ( p ) = 4 p µ . (A6)Finally, making use of Eq.(40) and Eqs.(A5)-(A6), onegets f (1 , µ = − ( im ) F (1 , µ (A7)and this accomplishes the proof. Appendix B: Another justification of Eq.(42)
In this appendix, we give another justification forEq.(42), which is based on the dynamics of virtual pro-cesses. To this end, we note a fact mentioned previously,that is, the term f (1) µ is for emergence and vanishing ofvirtual e -modes, while, f (2) µ is for virtual e -modes. Sincethe e -mode and e -mode in a virtual pair generated by anull A -mode have opposite momenta, a point p on therhs of Eq.(40a) should correspond to a point q = − p onthe rhs of Eq.(40b). This gives that f (1)0 + f (2)0 = 0 and f (1) i + f (2) i = 8 im Z d e qq i , i = 1 , , . (B1)To give further evaluation, we note the physical picturethat virtual pairs emerge randomly, such that there is nodifference between the probability for q and that for − q .This implies that the contribution of a value of q i on therhs of Eq.(B1) be canceled by that of − q i , resulting in avanishing result of the integration. [1] S. Weinberg, The Quantum Theory of Fields (CambridgeUniversity Press, Now York, 1996).[2] M.E. Peskin and D.V. Schroeder,
In Introduction to Quan- tum Field Theory (Westview Press, 1995).[3] C. Itzykson and J. B. Zuber,
Quantum Field Theory (McGraw-Hill, New York, 1980). [4] G. W. Mackey, Ann. Math. , 101 (1952); , 193 (1953);Acta. Math. , 265 (1958); Induced Representations ofGroups and Quantum Mechanics (Benjamin, New York,1968).[5] W. Pauli, Phys.Rev. , 716 (1940).[6] R.F. Streater and A.S. Wightman, PCT, Spin and Statis-tics, and All That (Benjamin/Cummings, Reading, Mass.,1964). [7] W.-g. Wang, Phys.Rev.A , 012112 (2016).[8] S.N. Gupta, Proceedings of Physical Society A , 681(1950).[9] As shown in a paper of Y. Gu, Phys. Rev. A88