aa r X i v : . [ h e p - t h ] M a y Solid quantization for non-point particles
P. Wang
1, 2 Institute of High Energy Physics, CAS, P. O. Box 918(4), Beijing 100049, China Theoretical Physics Center for Science Facilities, CAS, Beijing 100049, China
In quantum field theory, elemental particles are assumed to be point particles. As a result, theloop integrals are divergent in many cases. Regularization and renormalization are necessary inorder to get the physical finite results from the infinite, divergent loop integrations. We proposenew quantization conditions for non-point particles. With this solid quantization, divergence couldbe treated systematically. This method is useful for effective field theory which is on hadron degreesof freedom. The elemental particles could also be non-point ones. They can be studied in thisapproach as well.
Quantum field theory is the fundamental theory fornuclear and particle physics. The simplest way to quan-tize the field is to use canonical quantization which issimilar as in quantum mechanics. It is equivalent to thepath integral method. With the quantum field theory,one can study the micro process with Feynman rules.When do the high order calculation, the loop contribu-tion will appear. These integrals are often divergent, i.e.,they become infinite when momentum integration goesto infinity. This ultraviolet divergence is short-distancephenomenon.Many kinds of methods are introduced in quantumfield theory to deal with the divergence. One of the mostpopular method is dimensional regularization [1]. It pro-vides a systematic tool to obtain finite physical resultsfrom the infinity. Another is Pauli-Villars regularizationwhich adds fictitious particles to the theory with largemasses to cancel out the infinity [2].Quantum field theory with dimensional regularizationis very standard and widely accepted. It is also appliedin effective field theory which is on hadron degrees offreedom [3–5]. In hadron physics, there are a lot of phe-nomenological models where divergence is often treatedby adding a cutoff or form factor to the integral “byhand”. The cutoff or form factor can be related to thewave function which means hadrons are not point parti-cles [6, 7]. It can also be “derived” from the non-localinteraction [8, 9]. In other words, if particles are not pointones, there is no divergence appear from the beginning.There exists quantum field theories for point particles.Whether we can have some “theories” for non-point par-ticles in which divergence can be avoided systematically?We will show that from the new quantization conditions,one could get the modified propagators for non-point par-ticles. Divergence can be treated systematically.In fact, in the early 1950s, Yukawa has proposed thenon-local fields which described the non-point particles[10, 11]. It was assumed that the non-local field was afunction of four space-time operators x µ as well as of fourspace-time displacement operators p µ . Besides the nor-mal equation, this field satisfied another one which wasrelated to the radius of elemental particle. However, this idea did not get widely accepted because the aim to getrid of divergence was not easily established [12, 13]. InRefs. [14, 15], the authors claimed that there exists nomeaningful S matrix with non-local interaction. Whilesome authors pointed out that the violation of unitaryobserved in space/time noncommutative field theorieswas due to an improper definition of quantum field theoryon noncommutative spacetime (Quantum field theory onthe standard noncommutative spacetime is equivalent toa non-local theory on a commutative spacetime.) [16].As long as a proper perturbative setup is employed, non-local field theories may well be unitary in the sense thatprobabilities are always conserved. A proof of unitary of S matrix as well as causality in a non-local quantum fieldtheory has been shown in the paper of Alebastrov andEfimov [17, 18]. At the same time, non-local quantumelectrodynamics was widely discussed [19–24]. In recentyears, a lot of work has been done on the non-local phe-nomenological models as well as on the noncommutativefield theory [25–30]. For practise, one can use the uni-tary operator T exp { i R ∞−∞ d x L int ( x ) } , where L int ( x ) isthe non-local interaction, to do the perturbative expan-sion order by order [8, 31].In this paper, we propose new quantization conditionsfor non-point particles. Consistent with this solid quanti-zation, the non-local Lagrangian is straightforward. Dif-ferent from the traditional non-local case, here the freeLagrangian should be non-local as well.Let’s start with the traditional canonical quantizationfor the simplest scalar field. The traditional commutationrelations are:[ φ ( ~x, t ) , φ ( ~y, t )] = [ π ( ~x, t ) , π ( ~y, t )] = 0 , [ φ ( ~x, t ) , π ( ~y, t )] = iδ (3) ( ~x − ~y ) . (1)The δ function in the above equation means that a pointparticle and anti-particle can only be created at the sameposition point.The field and its conjugate partner can be expandedin momentum space, expressed as φ ( ~x, t ) = Z f dp (cid:2) a ( ~p ) e i~p · ~x − iω p t + a † ( ~p ) e − i~p · ~x + iω p t (cid:3) , (2) π ( ~x, t ) = Z f dp ( − i ) ω p (cid:2) a ( ~p ) e i~p · ~x − iω p t − a † ( ~p ) e − i~p · ~x + iω p t (cid:3) , (3)where f dp = d p (2 π ) ω p . (4)It is straightforward to obtain the commutation rela-tions between creation and annihilation operators:[ a ( ~p ) , a ( ~q )] = (cid:2) a † ( ~p ) , a † ( ~q ) (cid:3) = 0 , (cid:2) a ( ~p ) , a † ( ~q ) (cid:3) = (2 π ) ω p δ (3) ( ~p − ~q ) . (5)The creation operator creates a momentum state | p i = a † ( ~p ) | i which is normalized as Z f dp | p ih p | = 1 . (6)Because the particle is assumed to be point particle(behaves like δ function in position space), when ex-panded in momentum space, it has the same possibilityfor different momentum. However, the real particle couldbe like a wavepacket. It is partially localized in both po-sition and momentum space. The possibility of the parti-cle with high momentum is small. With high-momentumsuppression, the divergence in the loop integral may notappear.Therefore, we propose new quantization conditions(solid quantization):[ φ ( ~x, t ) , φ ( ~y, t )] = [ π ( ~x, t ) , π ( ~y, t )] = 0 , [ φ ( ~x, t ) , π ( ~y, t )] = i Φ ( ~x − ~y ) . (7)The function Φ ( ~x − ~y ) describes the correlation betweenfields at ~x and ~y . Due to the fact that particle is not adimensionless point particle, but a solid one, particles atdifferent positions could be partially superimposed whichmeans there exists some possibility that particle and an-tiparticle are created in different positions.One can also expand the field as Eq. (2) (In this case,we use capital letter A instead of a .) φ ( ~x, t ) = Z f dp (cid:2) A ( ~p ) e i~p · ~x − iω p t + A † ( ~p ) e − i~p · ~x + iω p t (cid:3) . (8)As a result, the creation and annihilation operators sat-isfy the following relations[ A ( ~p ) , A ( ~q )] = (cid:2) A † ( ~p ) , A † ( ~q ) (cid:3) = 0 , (cid:2) A ( ~p ) , A † ( ~q ) (cid:3) = (2 π ) ω p δ (3) ( ~p − ~q )Ψ( ~p ) . (9)Φ ( ~x ) and Ψ ( ~p ) obey the following relationsΦ ( ~x ) = Z d p (2 π ) Ψ( ~p )2 ( e i~p · ~x + e − i~p · ~x ) , (10) Ψ ( ~p ) = Z d x Φ( ~x )2 ( e i~p · ~x + e − i~p · ~x ) . (11)The above two equations generate two normalizationformulas Φ(0) = Z d p (2 π ) Ψ( ~p ) , (12)Ψ(0) = Z d x Φ( ~x ) = 1 . (13)Compared with the traditional commutation relationwhere Φ ( ~x ) = δ (3) ( ~x ), Φ ( ~x ) is normalized to be 1, whileΨ( ~p ) is normalized to be Φ(0).With the new quantization, the field can be written interms of traditional creation and annihilation operatorsas φ ( ~x, t ) = Z f dp p Ψ( ~p ) (cid:2) a ( ~p ) e i~p · ~x − iω p t + a † ( ~p ) e − i~p · ~x + iω p t (cid:3) . (14)It is easy to get the Feynman propagator of the scalarfield in the solid quantization. The propagator is definedas ∆ F ( x ′ − x ) = h | T φ ( x ′ ) φ ( x ) | i = Z f dk h θ ( t ′ − t ) e ik · ( x ′ − x ) + θ ( t − t ′ ) e − ik · ( x ′ − x ) i . (15)The integral expression of the step function is θ ( t ) = lim ǫ → + Z dτ πi e iτt τ − iǫ . (16)With the help of the above equation, the Feynman prop-agator can be obtained as∆ F ( x ′ − x ) = Z d k (2 π ) i Ψ( ~k ) e − ik · ( x ′ − x ) k − m + iǫ . (17)For the other fields, the quantization condition is simi-lar. For example, for spin 1/2 fermion, the nonzero anti-commutation relationship is (cid:8) ψ α ( ~x, t ) , ¯ ψ β ( ~y, t ) (cid:9) = γ αβ Φ( ~x − ~y ) . (18)Correspondingly, the field should be written as ψ ( ~x, t ) = X s = ± Z f dp p Ψ( ~p ) (cid:2) b s ( ~p ) u s ( ~p ) e i~p · ~x − iω p t + d † s ( ~p ) v s ( ~p ) e − i~p · ~x + iω p t (cid:3) , (19)where b and d † are normal annihilation and creation oper-ators. u s ( ~p ) and v s ( ~p ) are Dirac spinors. The propagatorof the spin 1/2 field can be obtained as S F ( x ′ − x ) = Z d k (2 π ) i Ψ( ~k )( k · γ + m ) e − ik · ( x ′ − x ) k − m + iǫ . (20)Vector field, say photon field can also be expanded as A µ ( ~x, t ) = X λ = ± Z f dp p Ψ( ~p ) (cid:2) a λ ( ~p ) ǫ µ ( ~p, λ ) e i~p · ~x − iω p t + a † λ ( ~p ) ǫ ∗ µ ( ~p, λ ) e − i~p · ~x + iω p t i , (21)where ǫ µ ( ~p, λ ) is the polarization vector. The photonpropagator can be written as D µνF ( x ′ − x ) = Z d k (2 π ) − i Ψ( ~k ) g µν e − ik · ( x ′ − x ) k − m + iǫ . (22)We should mention that, in principle, the functionΨ( ~p ) or Φ( ~x − ~y ) is particle dependent. It describes theparticle’s property in addition to the mass and width.Therefore, with the new quantization conditions, theFeynman rules should be changed correspondingly. Thenew propagator of the field is multiplied by a factor Ψ( ~k )and the external field is multiplied by a factor q Ψ( ~k ).A question may arise here that how to connect the newpropagator with the path integral formulation. The pathintegral for the free point-like field is defined as Z ( J ) = Z D φe i R d x [ L + Jφ ] , (23)where L = − ∂ µ φ∂ µ φ − m φ (24)is the Lagrangian density and J is the external current.For a solid particle, the free Lagrangian density is differ-ent. The density can be written as L = φ ( ∂ µ ∂ µ − m )2Ψ( i~∂ ) φ. (25)With the above Lagrangian density, the propagator ofscalar field obtained in the path integral formulation isthe same as that in solid canonical quantization. For thefermion and vector fields, the situation is the same.The factor Φ( ~x − ~y ) is the correlation of two parti-cles at ~x and ~y . If we choose Φ( ~x − ~y ) = δ (3) ( ~x − ~y ),Ψ( ~p ) will equal 1. All of the above propagators will bechanged back to the conventional ones. As we explainedpreviously, the particle could be a solid particle withthree space dimensions. The particle and antiparticlecan be created at small distance. Therefore, the functionof Φ( ~x − ~y ) can be a function which decreases with theincreasing distance | ~x − ~y | . The smaller the particle, thecloser the function to δ function.The above solid quantization provides the “kinemat-ics” for quantum field theory. Now let’s look at the “dy-namics” for non-point particles. Gauge invariance is afundamental method to get the strong or electro-weakinteractions. Since particles are not point ones, in gen-eral, the interaction among them is non-local. Similar as the non-local quark-meson interaction [8, 31], the gaugeinvariant interaction between fermion and gauge field,say photon field, can be written as L ( x ) = Z d a ¯ ψ ( t, ~x + ~a e iI ( ~x + ~a/ ,~x ) γ µ iD µ e − iI ( ~x − ~a/ ,~x ) ψ ( t, ~x − ~a F ( ~a ) , (26)where D µ = ∂ µ − ig R d bA µ ( t, ~x + ~b ) G ( ~a,~b ) and I ( y, x ) = g R yx dz µ R d bA µ ( z , ~z + ~b ) G ( ~a,~b ). g is the coupling con-stant. The function F ( ~a ) and G ( ~a,~b ) are related to thesize of fermion and gauge fields. The non-local couplingdepends on the distance between the two fermion fieldsand the distance between gauge field and the center oftwo fermion fields. The coupling G ( ~a,~b ) can be factor-ized as Φ( ~a )Φ g ( ~b ) F ( ~a ) , where Φ( ~a ) is the correlation functionbetween two fermions at distance ~a defined in Eq. (7).Φ g ( ~b ) the correlation function for gauge fields. The par-ticular choice of G ( ~a,~b ) is to get the interacting term R d a R d b ¯ ψ ( t, ~x + ~a ) γ µ ψ ( t, ~x − ~a ) A µ ( t, ~x + ~b )Φ( ~a )Φ g ( ~b )which provides the possibility Φ( ~a )Φ g ( ~b ) for the non-localinteraction.The above Lagrangian is invariant under the followinggauge transformation: ψ ( t, ~x − ~a/ → e ig eff θ ( t,~x − ~a/ ψ ( t, ~x − ~a/ , ¯ ψ ( t, ~x + ~a/ → ¯ ψ ( t, ~x + ~a/ e − ig eff θ ( t,~x + ~a/ ,A µ ( t, ~x + ~b ) → A µ ( t, ~x + ~b ) + ∂ µ θ ′ ( t, ~x + ~b ) , (27)where g eff is defined as Φ( ~a )Φ g ( ~b ) F ( ~a ) which can be understoodas the effective charge of a non-local electromagnetic cur-rent with a distance ~a between a fermion and an anti-fermion. The appearance of F ( ~a ) in the denominator of g eff is because of the non-point property of the fermions.The functions θ and θ ′ have the following relation θ ( t, ~x ) = Z d bθ ′ ( t, ~x + ~b ) . (28)The strong and weak interaction can be easily obtainedin the same way with SU (3) and SU (2) generators.The free Lagrangian density without gauge field is L ( x ) = Z d a ¯ ψ ( t, ~x + ~a γ µ i∂ µ ψ ( t, ~x − ~a F ( ~a ) . (29)This non-local free Lagrangian is different from that inthe traditional non-local models where the free part of theLagrangian is local and the interaction part is a non-localcoupling of point particles [8, 25–28, 31]. Our treatmentis more consistent. Due to the non-point assumption,the non-local Lagrangian is straightforward and it is alsonecessary because of the solid quantization. After movingthe position to the same point by the translation opera-tor, it is straightforward to rewrite the above Lagrangianas L ( x ) = ¯ ψ ( x ) γ µ i∂ µ ˜ F ( i~∂ ) ψ ( x ) , (30)where ˜ F ( i~∂ ) is the Fourier transformation of of F ( ~a ), i.e.,˜ F ( i~∂ ) = Z d ae i~a · i~∂ F ( ~a ) . (31)Comparing Eqs. (25) and (30), we can get the relation-ship ˜ F ( i~∂ ) = 1 / Ψ( i~∂ ). One can see that the solid quanti-zation is consistent with the path integral approach withnon-local Lagrangian density.The interaction term is written as L int ( x ) = g Z d a Z d b ¯ ψ ( t, ~x + ~a e iI ( ~x + ~a/ ,~x ) γ µ A µ ( t, ~x + ~b ) e − iI ( ~x − ~a/ ,~x ) ψ ( t, ~x − ~a ~a )Φ g ( ~b )+ Z d a ¯ ψ ( t, ~x + ~a e iI ( ~x + ~a/ ,~x ) − γ µ i∂ µ e − iI ( ~x − ~a/ ,~x ) ψ ( t, ~x − ~a F ( ~a )+ Z d a ¯ ψ ( t, ~x + ~a γ µ i∂ µ ( e − iI ( ~x − ~a/ ,~x ) − ψ ( t, ~x − ~a F ( ~a ) . (32)The field can be expanded in power of ~a and ~b as ψ ( t, ~x + ~a ) = ψ ( t, ~x ) + ~∂ψ ( t, ~x ) · ~a + O ( ~a ) ,A µ ( t, ~x + ~b ) = A µ ( t, ~x ) + ~∂A µ ( t, ~x ) · ~b + O ( ~b ) . (33)The interaction can be expressed as L int ( x ) = g ¯ ψ ( x ) γ µ A µ ( x ) ψ ( x ) + O (¯ ~a, ¯ ~b ) , (34)where ¯ ~a and ¯ ~b reflect the size of the particles defined as ~a = Z d a ~a Φ( ~a ) ,~b = Z d b ~b Φ g ( ~b ) . (35)For the “free” Lagrangian, we should not expand itin terms of ~a . The “free” Lagrangian provides propaga-tors for solid particles. The further volume effect of solidparticles can be added order by order. If the particle’ssize is small enough, we can neglect high order terms inEq. (34). The lowest order interaction term is the sameas that in the local case.The solid quantization is valid for elemental particlesas well as for hadrons if elemental particles are not pointones either. With the new propagator, the loop integra-tion is convergent. For example, let’s look at the follow-ing integration which appears in the photon self-energyat one-loop level: I = Z d k (2 π ) Ψ( ~k )Ψ( ~k + ~p )[ k − m ] [( p + k ) − m ] , (36) where p is the external momentum of photon. k and k + p are the internal momentum of two electron or quarkpropagators. After integration of k , the above equationcan be written as I = Z d k π ) − i Ψ( ~k )Ψ( ~k + ~p ) ω ( ~k ) h ( ω ( ~k ) + ω ( ~p )) − ω ( ~k + ~p ) i − i Ψ( ~k )Ψ( ~k + ~p ) ω ( ~k + ~p ) h ( ω ( ~k + ~p ) − ω ( ~p )) − ω ( ~k ) i , (37)where ω ( ~q ) = p ~q + m . Without the factor Ψ( ~k ) andΨ( ~k + ~p ), the above integration is log-divergent. Sincethe particle is a solid one with three dimensions, its wave-function is suppressed at high momentum. If we chooseΨ( ~k ) to be a dipole or Gauss function, the integration isconvergent.Without renormalization, the running coupling con-stant can also be understood. With the new quantiza-tion conditions, even at tree level, the coupling constantwill be associated with a momentum dependent factor.For example, for the fermion-boson coupling, if the ini-tial and final momentum of fermions are − ~q/ ~q/ ~q , and the momentumdependent factor of the coupling constant at tree levelis p Ψ f ( − ~q/ f ( ~q/ g ( ~q ). The labels f and g are forfermion and gauge boson, respectively. The asymptoticfree is a general property not only for strong interaction.It is because the particle is not a point one. The momen-tum is partially localized which favors at low value.Investigating quantum electrodynamic process is agood and clean way to test this quantization for elemen-tal particles. Let’s study the electron-photon Comptonscattering for an example: e − ( p, s ) + γ ( k, ǫ ) → e − ( p ′ , s ′ ) + γ ( k ′ , ǫ ′ ) , (38)where p and p ′ , k and k ′ are the initial and final mo-mentum of electron and photon, respectively. s and ǫ aretheir spin and polarization. The scattering amplitudecan be obtained as M = − e q Ψ e ( ~p )Ψ e ( ~p ′ )Ψ γ ( ~k )Ψ γ ( ~k ′ )¯ u s ′ ( ~p ′ ) h Ψ e ( ~p + ~k ) ǫ ′∗ p + k − m ǫ + Ψ e ( ~p − ~k ′ ) ǫ p − 6 k ′ − m ǫ ′∗ (cid:21) u s ( ~p ) . (39)In the Lab frame where the initial electron is at rest,after summing over the initial and final electron spins,averaged square of amplitude is simplified as¯ M = e m (cid:20) Aω + Bω ′ + Cωω ′ (cid:21) , (40)where ω and ω ′ are energy of initial and final photon. A , B and C is expressed as A = 8 mω (cid:2) k · ǫ ′ ) + mω ′ (cid:3) F ( ω, ω ′ ) , (41) B = − mω ′ (cid:2) k ′ · ǫ ) − mω (cid:3) F ( ω, ω ′ ) , (42) C = (cid:2) m ωω ′ [2( ǫ · ǫ ′ ) − − mω ′ ( k · ǫ ′ ) +16 mω ( k ′ · ǫ ) (cid:3) F ( ω, ω ′ ) , (43)where F ( ω, ω ′ ) = Ψ e ( ω + ω ′ − ωω ′ cosθ )Ψ γ ( ω )Ψ γ ( ω ′ )Ψ e ( ω ) ,F ( ω, ω ′ ) = Ψ e ( ω + ω ′ − ωω ′ cosθ )Ψ γ ( ω )Ψ γ ( ω ′ )Ψ e ( ω ′ ) ,F ( ω, ω ′ ) = Ψ e ( ω + ω ′ − ωω ′ cosθ )Ψ γ ( ω )Ψ γ ( ω ′ )Ψ e ( ω )Ψ e ( ω ′ ) (44)are the additional functions associated with the newquantization and θ is the angle between initial and finalmomentum of photon. With α = e / π , the differentialcross section can then be written as dσd Ω = α m (cid:18) ω ′ ω (cid:19) (cid:20) Aω + Bω ′ + Cωω ′ (cid:21) . (45)In the point particle approximation, the function of Ψ( ~p )equals 1 and the above cross section is changed back tothe traditional one dσd Ω = α m (cid:18) ω ′ ω (cid:19) (cid:20) ω ′ ω + ωω ′ + 4( ǫ · ǫ ′ ) − (cid:21) . (46)To test the solid quantization, it is interesting to mea-sure cross section of high energy electron-photon Comp-ton scattering because the function Ψ e and Ψ γ will havesignificant decrease at high momentum (energy). Thesmaller the particle, the larger the energy at which crosssection has a clear difference from the traditional one.Due to the inclusion of the size of the particle, theabove fields ( φ ( x ) or ψ ( x )) as well as the propagators arenot Lorentz covariant quantities. Our start point is thatat each time t , each particle has a distribution on space.It is obviously non-relativistic though this physical pic-ture is very clear and similar to many phenomenologicalmodels. It is also easy for us to apply this approach innumerical calculation. For example, in the effective fieldtheory, finite regularization in which a ~k dependent reg-ulator u ( ~k ) was introduced “by hand” was used to getrid of the divergence [32, 33]. We can use the abovesolid propagators for hadrons to investigate the mesonloop contribution. Compared with finite range regular-ization, this approach automatically gives the “regulator”for each diagram. The obtained “regulator” is diagramdependent.Now we give the relativistic version of the solid quanti-zation. Different from the non-relativistic case, the fieldhas a distribution on four dimensional space-time. For ascalar field φ ( x ), it can be written as φ ( x ) = Z d p (2 π ) H ( p ) (cid:2) α p e − ip · x + α † p e ip · x (cid:3) . (47) The operators α p and α † p have the following commutationrelations: [ α p , α q ] = (cid:2) α † p , α † q (cid:3) = 0 , (cid:2) α p , α † q (cid:3) = (2 π ) δ (4) ( p − q ) . (48)The commutation relations of scalar field and its conju-gate are[ φ ( ~x, t ) , π ( ~y, t )] = Z d p (2 π ) H ( p ) ip ( e i~p · ~x + e − i~p · ~x )= Z d p (2 π ) i Ψ( ~p )2 ( e i~p · ~x + e − i~p · ~x ) ≡ i Φ( ~x − ~y ) , (49)where Ψ( ~p ) = Z dp π H ( p ) p . (50)For point particle with mass m , Ψ( ~p ) = 1 and H ( p ) =2 πδ ( p − m ). We should mention that H ( p ) is pro-portional to δ / ( p − m ) instead of δ ( p − m ). Thisis because the field is expanded in terms of α p and α † p instead of a p and a † p .For simplicity, we rewrite the scalar field as φ ( x ) = Z d p (2 π ) dM H ( M ) δ ( p − M ) (cid:2) α p e − ip · x + α † p e ip · x (cid:3) = Z d p (2 π ) ω M dM H ( M ) h α ~p,ω M e i~p · ~x − iω M t + α † ~p,ω M e − i~p · ~x + iω M t i , (51)where ω M = p ~p + M .We can get the propagator of scalar field as∆ F ( x ′ − x ) = Z d k π ) ω M ω M ′ dM dM ′ H ( M ) H ( M ′ ) δ ( ω M ′ − ω M ) h θ ( t ′ − t ) e ik · ( x ′ − x ) + θ ( t − t ′ ) e ik · ( x − x ′ ) i , (52)where δ ( ω M ′ − ω M ) = 2 ω M δ ( M ′ − M ). With the defi-nition of θ function, the propagator can be written as∆ F ( x ′ − x ) = Z d k (2 π ) dM π iH ( M ) k − M + iǫ e − ik · ( x ′ − x ) . (53)Again, if H ( M ) = 2 πδ ( M − m ), the propagator is thesame as that for point particle with mass m . If H ( M )is chosen to be 2 π [ δ ( M − m ) − δ ( M − Λ )], one canget Pauli-Villars regularization.In the relativistic case, the Lagrangian density can bewritten in the same way as Eq. (26) except the integralis on four dimensional space-time because both time andspace are non-local, i.e. L ( x ) = Z d a ¯ ψ ( x + a e iI ( x + a/ ,x ) γ µ iD µ e − iI ( x − a/ ,x ) ψ ( x − a F ( a ) , (54)where D µ = ∂ µ − ig R d bA µ ( x + b ) Φ( a )Φ g ( b ) F ( a ) and I ( y, x ) = g R yx dz µ R d bA µ ( z + b ) Φ( a )Φ g ( b ) F ( a ) . In the relativistic case,the function Φ( a ) or Φ g ( b ) is defined to be the Fouriertransformation of 1 / ˜ F ( p ), i.e.Φ( a ) = Z d p (2 π ) F ( p ) e ip · a , (55)where ˜ F ( p ) is the Fourier transformation of function F ( a ). Similar as in the non-relativistic case, by compar-ing with the propagators obtained in solid quantizationand path integral methods, one can get the relationshipbetween H ( p ) and ˜ F ( p ): Z dM π H ( M ) p − M = 1˜ F ( p )( p − m ) . (56)Again, one can see that, to be consistent with the solidquantization, the Lagrangian is non-local in both free andinteraction parts.Due to the non-local property, the conversation lawsare modified accordingly [34]. The currents or chargesare not conserved at any space-time point. But theintegral of them are conserved. For example, in thenon-local case, there exists no unitary time evolutionoperator U ( t , t ) for given t and t . But there ex-ists a unitary time evolution operator U ( −∞ , ∞ ) ≡ T exp { i R ∞−∞ d x L int ( x ) } . This can be easily understoodsince a fermion, an anti-fermion and a gauge field can beannihilated/created at different time. The possibility ofstate is not conserved at a fixed time. But the integral ofthe possibility over the time is conserved. In other words,the time evolution operator U ( −∞ , ∞ ) is unitary.Though the relativistic version of the solid quantiza-tion is Lorentz invariant, the causality condition is dif-ferent from the traditional quantum field theory. Forexample for scalar field in local case, the equal-time com-mutator [ φ ( ~x ) , π ( ~y )] equal zero if x and y are spacelikeseparated, i.e. ( x − y ) − ( ~x − ~y ) <
0. In non-localcase, there exists some possibility of non-zero commuta-tor [ φ ( ~x ) , π ( ~y )] for − ( ~x − ~y ) <
0. The non-zero com-mutator is because of the space-time distribution of thenon-point fields. Therefore, the classic causality condi-tion turns into a quantum (possibility) condition. Ap-proximately, one may think the two non-point fields arespacelike separated if ( x − y ) − ( ~x − ~y ) < − ~a , where ~a is the size of the field.In summary, we have proposed a new quantization -solid quantization for non-point fields. The divergence in the loop integrals for point particles needs to be takencare of with the regularization method. This solid quan-tization condition is very natural and based on the ideathat a physical particle is not a mathematic point one.The function in the commutation relations is another fun-damental properties of the particle as well as mass, spin,width, etc. The divergence of loop integration could besystematically avoided from the beginning. Both non-relativistic and relativistic version of this solid quantiza-tion are given.For the dimensional regularization, one has to use infi-nite Lagrangian or bare Lagrangian to get finite physicalresults. This method provide an interesting approachwhich is quite different from traditional quantum fieldtheory. In this paper, we did not specify the functionof Φ( ~x ), Ψ( ~p ) or H ( p ). To get more information aboutthe function of the particle, it is important to do furthernumerical calculations to compare with the experimentsand traditional results. Acknowledgements
The author would like to thank Prof. Y. B. Dong forhelpful discussions. [1] Gerard ’t Hooft, M.J.G. Veltman, Nucl. Phys. B , 189(1972).[2] W. Pauli, F. Villars, Rev. Mod. Phys. , 434 (1949).[3] J. Gasser and H. Leutwyler, Nucl. Phys. B , 465(1985).[4] G. Ecker, Prog. Part. Nucl. Phys. , 1 (1995).[5] B. Kubis and U. Meissner, Eur. Phys. J. C , 747 (2001).[6] D. H. Lu, A. W. Thomas and A. G. Williams, Phys. Rev.C , 2628 (1998).[7] V. E. Lyubovitskij, P. Wang, T. Gutsche and A. Faessler,Phys. Rev. C , 055204 (2002).[8] M.A. Ivanov and V.E. Lyubovitskij, Phys. Lett. B ,435 (1997).[9] T. Hell, K. Kashiwa and W. Weise, Phys. Rev. D ,114008 (2011).[10] H. Yukawa, Phys. Rev. , 219 (1950).[11] H. Yukawa, Phys. Rev. , 1047 (1950).[12] R. Marnelius, Phys. Rev. D , 2472 (1973).[13] C. Moller, in Proceedings of the International Conferenceon Theoretical Physics, Kyoto and Tokyo, September,1953 (Science Council of Japan, Tokyo, 1954).[14] R. Marnelius, Phys. Rev. D , 3411 (1974).[15] T. Imamura, S. Sunakawa and R. Utiyama, Prog. Theor.Phys. , 291 (1954).[16] D. Bahns, S. Doplicher, K. Fredenhagen and G. Piacitelli,Phys. Lett. B , 178 (2002).[17] V. A. Alebastrov, G. V. Efimov, Commun. Math. Phys. , 1 (1973).[18] V. A. Alebastrov, G. V. Efimov, Commun. Math. Phys. , 11 (1974).[19] G. V. Efimov, S. Z. Seltser, Annals Phys. , 124 (1971). [20] G. V. Efimov, Annals Phys. , 466 (1972).[21] G. V. Efimov, O. A. Mogilevsky, Nucl. Phys. B44 , 541(1972).[22] G. V. Efimov, M. A. Ivanov, O. A. Mogilevsky, AnnalsPhys. , 169 (1977).[23] T. U. Phat, Acta Phys. Polon. B4 , 311 (1973).[24] J. Terning, Phys. Rev. D , 887 (1991).[25] A. E. Radzhabov, D. Blaschke, M. Buballa, and M. K.Volkov, Phys. Rev. D , 1702 (2011).[27] S. Noguera and N. N. Scoccola, Phys. Rev. D , 114002(2008).[28] A. Faessler, T. Gutsche, M. A. Ivanov, J. G. Korner and Valery E. Lyubovitskij, Phys. Rev. D , 034025 (2009).[29] A. Armoni, arXiv:1107.3651.[30] A. P. Balachandran, A. Ibort, G. Marmo and M. Mar-tone, JHEP , 057 (2011).[31] A. Faessler, T. Gutsche, M. A. Ivanov, V. E. Lyubovitskijand P. Wang, Phys. Rev. D , 014011 (2003).[32] D. B. Leinweber, A. W. Thomas and R. D. Young, Phys.Rev. Lett. , 242002 (2004).[33] P. Wang, D. B. Leinweber, A. W. Thomas, and R. D.Young, Phys. Rev. D , 094001 (2009).[34] W. Garczynski and J. Stelmach, Acta Phys. Polon. B15