aa r X i v : . [ phy s i c s . g e n - ph ] O c t A New Case for Direct Action
Michael IbisonInstitute for Advanced Studies at Austin,11855 Research Boulevard, Austin, TX 78759, USAemail: [email protected]
An obstacle to the development of direct action version of electromag-netism was that in the end it failed to fulfill its initial promise of avoidingthe problem of infinite Coulomb self-energy in the Maxwell theory of theclassical point charge. This paper suggests a small but significant modifica-tion of the traditional direct action theory which overcomes that obstacle.Self-action is retained but the associated energy is rendered finite and equalto zero in the special case of null motion.
Arguably there are three essential characteristics peculiar to EM direct action that dis-tinguish it from the Maxwell field-theoretic approach:i) The (initial) promise of zero self-energy.ii) EM time symmetry, and hence the need to find an explanation for the observedasymmetry of radiation outside of EM.iii) No vacuum degrees of freedom, hence no EM second-quantization and no EM ZPF.The focus below is exclusively on i). Wheeler and Feynman [1, 2] attempted to resolveii) though it was later found that the cosmological boundary condition - identified asnecessary in order to explain the predominance of retarded radiation - could not be metby any reasonable cosmology [3, 4]. In any case their argument has since been criticizedon the grounds that it hides the assumption of a thermodynamic arrow of time uponwhich it bases a derivation of the EM arrow of time [5, 6]. An alternative suggestion [7]remains unproven. Davies [8, 9, 10] has shown that the results of relativistic QM are1 ichael Ibison A New Case for Direct Actionichael Ibison A New Case for Direct Action
Arguably there are three essential characteristics peculiar to EM direct action that dis-tinguish it from the Maxwell field-theoretic approach:i) The (initial) promise of zero self-energy.ii) EM time symmetry, and hence the need to find an explanation for the observedasymmetry of radiation outside of EM.iii) No vacuum degrees of freedom, hence no EM second-quantization and no EM ZPF.The focus below is exclusively on i). Wheeler and Feynman [1, 2] attempted to resolveii) though it was later found that the cosmological boundary condition - identified asnecessary in order to explain the predominance of retarded radiation - could not be metby any reasonable cosmology [3, 4]. In any case their argument has since been criticizedon the grounds that it hides the assumption of a thermodynamic arrow of time uponwhich it bases a derivation of the EM arrow of time [5, 6]. An alternative suggestion [7]remains unproven. Davies [8, 9, 10] has shown that the results of relativistic QM are1 ichael Ibison A New Case for Direct Actionichael Ibison A New Case for Direct Action not affected by iii) provided all radiation is absorbed. Boyer [11] amongst others hasshown that some of behaviours attributed to the EM ZPF of QED such as the Lambshift and the Casimir effect can be treated within stochastic electrodynamics (SED) -a variant of classical EM augmented by the presence of a (classical) Lorentz-Invariantbackground field. The direct action paradigm is that the EM potentials are not independent degrees offreedom apart from the currents that source them. If the field theoretic form of theclassical action is taken to be I = Z d x (cid:26) A ◦ (cid:0) ∂ A (cid:1) + κ ( ∂ ◦ A ) − A ◦ j (cid:27) − X q m q Z q dx q (1)then variation of A would normally give ∂ A = j in the Lorenz ( κ = 0) gauge with theimplication that A = G ∗ j + A cf ; ∂ G = δ , ∂ A cf = 0 for some (scalar) Green’sfunction G appropriate to the boundary conditions. By contrast direct action maintainsthat A cf = 0 for some G that can be expressed as independent of j : no fields exist thatcannot be explained by currents. Implementing this in (1) gives I = − Z d x j ◦ G ∗ j − m q Z q dx q . (2)Explicating the currents as j ( x ) = X q e q Z dx q ( λ ) δ ( x − x q ( λ ))the action becomes I = − X p,q e p e q Z dx p ( κ ) ◦ Z dx q ( λ ) G ( x p ( κ ) − x q ( λ )) − X q m q Z q dx q . (3) That which is generally taken to be retarded radiation is then really an interaction between sources. The Casimir analysis in Itzykson and Zuber [12] for example is essentially classical and equal to thatof SED. A and j are Lorentz vectors with suppressed indexes. The symbol ◦ denotes a Lorentz scalar product sothat A ◦ j = A a j a = φρ − A . j , for example, where A . j is the Euclidean scalar product. dx = dx ◦ dx . q is a particle index. Everywhere c = 1. The symbol ∗ denotes convolution so that G ∗ j = R d x ′ G ( x − x ′ ) j ( x ′ ), δ is the Dirac delta function. x and x q ( λ ) are both 4-vectors. λ parameterizes the world line. It is not necessarily a Lorentz scalar. δ ( x − x q ( λ )) is shorthand for Q a =0 δ ` x a − x aq ( λ ) ´ where a indexes the components of a Lorentzvector. Notice there is no gauge freedom in (3). ichael Ibison A New Case for Direct Actionichael Ibison A New Case for Direct Action
Arguably there are three essential characteristics peculiar to EM direct action that dis-tinguish it from the Maxwell field-theoretic approach:i) The (initial) promise of zero self-energy.ii) EM time symmetry, and hence the need to find an explanation for the observedasymmetry of radiation outside of EM.iii) No vacuum degrees of freedom, hence no EM second-quantization and no EM ZPF.The focus below is exclusively on i). Wheeler and Feynman [1, 2] attempted to resolveii) though it was later found that the cosmological boundary condition - identified asnecessary in order to explain the predominance of retarded radiation - could not be metby any reasonable cosmology [3, 4]. In any case their argument has since been criticizedon the grounds that it hides the assumption of a thermodynamic arrow of time uponwhich it bases a derivation of the EM arrow of time [5, 6]. An alternative suggestion [7]remains unproven. Davies [8, 9, 10] has shown that the results of relativistic QM are1 ichael Ibison A New Case for Direct Actionichael Ibison A New Case for Direct Action not affected by iii) provided all radiation is absorbed. Boyer [11] amongst others hasshown that some of behaviours attributed to the EM ZPF of QED such as the Lambshift and the Casimir effect can be treated within stochastic electrodynamics (SED) -a variant of classical EM augmented by the presence of a (classical) Lorentz-Invariantbackground field. The direct action paradigm is that the EM potentials are not independent degrees offreedom apart from the currents that source them. If the field theoretic form of theclassical action is taken to be I = Z d x (cid:26) A ◦ (cid:0) ∂ A (cid:1) + κ ( ∂ ◦ A ) − A ◦ j (cid:27) − X q m q Z q dx q (1)then variation of A would normally give ∂ A = j in the Lorenz ( κ = 0) gauge with theimplication that A = G ∗ j + A cf ; ∂ G = δ , ∂ A cf = 0 for some (scalar) Green’sfunction G appropriate to the boundary conditions. By contrast direct action maintainsthat A cf = 0 for some G that can be expressed as independent of j : no fields exist thatcannot be explained by currents. Implementing this in (1) gives I = − Z d x j ◦ G ∗ j − m q Z q dx q . (2)Explicating the currents as j ( x ) = X q e q Z dx q ( λ ) δ ( x − x q ( λ ))the action becomes I = − X p,q e p e q Z dx p ( κ ) ◦ Z dx q ( λ ) G ( x p ( κ ) − x q ( λ )) − X q m q Z q dx q . (3) That which is generally taken to be retarded radiation is then really an interaction between sources. The Casimir analysis in Itzykson and Zuber [12] for example is essentially classical and equal to thatof SED. A and j are Lorentz vectors with suppressed indexes. The symbol ◦ denotes a Lorentz scalar product sothat A ◦ j = A a j a = φρ − A . j , for example, where A . j is the Euclidean scalar product. dx = dx ◦ dx . q is a particle index. Everywhere c = 1. The symbol ∗ denotes convolution so that G ∗ j = R d x ′ G ( x − x ′ ) j ( x ′ ), δ is the Dirac delta function. x and x q ( λ ) are both 4-vectors. λ parameterizes the world line. It is not necessarily a Lorentz scalar. δ ( x − x q ( λ )) is shorthand for Q a =0 δ ` x a − x aq ( λ ) ´ where a indexes the components of a Lorentzvector. Notice there is no gauge freedom in (3). ichael Ibison A New Case for Direct Actionichael Ibison A New Case for Direct Action The traditional choice in direct action is the symmetric propagator of advanced andretarded influences G ( x ) = δ (cid:0) x (cid:1) / π . (Any anti-symmetric component, even if per-mitted, makes no contribution to the action due to the permutation symmetry of theparticles.) With this and explicit parameterization the action becomes I = − π X p,q e p e q Z dκ Z dλ ˙ x p ( κ ) ◦ ˙ x q ( λ ) δ (cid:16) ( x p ( κ ) − x q ( λ )) (cid:17) − X q m q Z q dx q . (4) The total action (4) for N particles can be written as I = X p,qp = q I p,q + N I self − X q m q Z q dx q (5)where I self is the electromagnetic self-action I self = − e π lim ε → X σ = ± w ( σ ) Z dκ Z dλ ˙ x ( κ ) ◦ ˙ x ( λ ) δ (cid:16) ( x ( κ ) − x ( λ )) − σε (cid:17) where the Lorentz scalar ε is a small regularization parameter - sufficiently small tocapture only the local light-cone self-intersections of the world line. σ = ± w ( σ ) is a dimensionless weightof order unity. The limit ε → κ = λ : I self = − e π lim ε → X σ = ± w ( σ ) Z dξ Z dλ ˙ x ( λ + ξ ) ◦ ˙ x ( λ ) δ (cid:16) ( x ( λ + ξ ) − x ( λ )) − σε (cid:17) = − e π lim ε → X σ = ± w ( σ ) Z dλ (cid:0) ˙ x ( λ ) + O ( ξ ) (cid:1) Z dξδ (cid:0) ξ ˙ x ( λ ) + O (cid:0) ξ (cid:1) − σε (cid:1) = − e π lim ε → X σ = ± w ( σ ) Z dλ sgn (cid:0) ˙ x ( λ ) (cid:1) | ξ | (cid:12)(cid:12)(cid:12)(cid:12) ξ x λ )= σε + O (cid:0) ξ (cid:1)! = − lim ε → e π | ε | Z dλ (cid:16) w (1) Θ (cid:0) ˙ x ( λ ) (cid:1) p ˙ x ( λ ) − w ( −
1) Θ (cid:0) − ˙ x ( λ ) (cid:1) p − ˙ x ( λ ) (cid:17) + O (cid:0) ε (cid:1) . (6) Θ ( x ) is the Heaviside step function. Its value exactly at x = 0 is immaterial here since exactlylight-speed motion is excluded. ichael Ibison A New Case for Direct Actionichael Ibison A New Case for Direct Action
1) Θ (cid:0) − ˙ x ( λ ) (cid:1) p − ˙ x ( λ ) (cid:17) + O (cid:0) ε (cid:1) . (6) Θ ( x ) is the Heaviside step function. Its value exactly at x = 0 is immaterial here since exactlylight-speed motion is excluded. ichael Ibison A New Case for Direct Actionichael Ibison A New Case for Direct Action Note that the above is invalid if the world line is exactly null. Contribution to theaction at order O (cid:0) ε (cid:1) is ignored since the primary interest here is to render the self-energy finite in the limit. Specializing in this paper to time-monotonic world lines theintegration parameter can be set to the laboratory time λ = x ( λ ) = tI self = Z dt L self ( v ( t )) L self = − lim ε → e π | ε | (cid:16) w (1) Θ (cid:0) − v (cid:1) p − v − w ( −
1) Θ (cid:0) v − (cid:1) p v − (cid:17) (7)for which the canonical self-energy is H self = v . ∂L self ∂ v − L self = lim ε → e π | ε | " w (1) Θ (cid:0) − v (cid:1) √ − v + w ( −
1) Θ (cid:0) v − (cid:1) √ v − (8)provided v = 1. The usual singular result for the electromagnetic self-energy is nowevident as ε → m bare / √ − v , the latter comingfrom the third term in (5). The corresponding regularizing factors are w (1) = 1and w ( −
1) = 0. The method outlined here is a Lorentz-Invariant generalization of thefamiliar method employing a small charged shell whose radius goes to zero. The energy(8) as a function of speed is given in figure 1.
An early attraction of direct action was that it permitted the easy elimination of selfenergy by excluding self-action from the action Schwarzschild [14], Tetrode [15], andFokker [16]. In (4) P p,q → P p,q ; p = q and therefore H self = L self = 0. Later it was re-alized that self-action generates physically observable consequences and so cannot besimply removed - at least from QED. Feynman [17] for example noted that pair creationcan be regarded as a form of promoted self-action. One of the alleged advantages ofdirect action (over field theory) was then lost. Time reversals, which necessitate that at least some part of the trajectory is spacelike, were implicitlyexcluded at the outset by the particular form of the mechanical action. They are not excluded bythe electromagnetic part of the action alone however. Use has been made of the condition that the world line is not null, whereupon the derivatives of theHeaviside functions vanish. A second contribution to mass renormalization comes from the ZPF in QED and stochastic electro-dynamics. Notice in this derivation the absence of a problem associated with a factor of 4/3 destroying Lorentzcovariance - see [13] for a nice review of the history of that problem. ichael Ibison A New Case for Direct Actionichael Ibison A New Case for Direct Action
An early attraction of direct action was that it permitted the easy elimination of selfenergy by excluding self-action from the action Schwarzschild [14], Tetrode [15], andFokker [16]. In (4) P p,q → P p,q ; p = q and therefore H self = L self = 0. Later it was re-alized that self-action generates physically observable consequences and so cannot besimply removed - at least from QED. Feynman [17] for example noted that pair creationcan be regarded as a form of promoted self-action. One of the alleged advantages ofdirect action (over field theory) was then lost. Time reversals, which necessitate that at least some part of the trajectory is spacelike, were implicitlyexcluded at the outset by the particular form of the mechanical action. They are not excluded bythe electromagnetic part of the action alone however. Use has been made of the condition that the world line is not null, whereupon the derivatives of theHeaviside functions vanish. A second contribution to mass renormalization comes from the ZPF in QED and stochastic electro-dynamics. Notice in this derivation the absence of a problem associated with a factor of 4/3 destroying Lorentzcovariance - see [13] for a nice review of the history of that problem. ichael Ibison A New Case for Direct Actionichael Ibison A New Case for Direct Action Figure 1: Regularized Maxwell self-energy as function of speed. All magnitudes → ∞ as ε → w ( σ ) = σ , so the totalaction is I = − π X σ = ± X p,q e p e q w ( σ ) Z dκ Z dλ ˙ x p ( κ ) ◦ ˙ x q ( λ ) δ (cid:16) ( x p ( κ ) − x q ( λ )) − σε (cid:17) − X q m q Z q dx q , (9)the self action is L self = − lim ε → e π | ε | (cid:16) Θ (cid:0) − v (cid:1) p − v + Θ (cid:0) v − (cid:1) p v − (cid:17) (10)and the self-energy H self = v . ∂L self ∂ v − L self = lim ε → e π | ε | " Θ (cid:0) − v (cid:1) √ − v − Θ (cid:0) v − (cid:1) √ v − (11)is plotted in figure 2. This implies the energy is zero at light speed and is infinitein magnitude at all other speeds (positive infinite at sub-luminal speeds, negative infi-nite at superluminal speeds). The steps leading to (8) are not valid precisely at lightspeed however. An interpretation consistent with (10) is that the charge is comprised5 ichael Ibison A New Case for Direct Actionichael Ibison A New Case for Direct Action
An early attraction of direct action was that it permitted the easy elimination of selfenergy by excluding self-action from the action Schwarzschild [14], Tetrode [15], andFokker [16]. In (4) P p,q → P p,q ; p = q and therefore H self = L self = 0. Later it was re-alized that self-action generates physically observable consequences and so cannot besimply removed - at least from QED. Feynman [17] for example noted that pair creationcan be regarded as a form of promoted self-action. One of the alleged advantages ofdirect action (over field theory) was then lost. Time reversals, which necessitate that at least some part of the trajectory is spacelike, were implicitlyexcluded at the outset by the particular form of the mechanical action. They are not excluded bythe electromagnetic part of the action alone however. Use has been made of the condition that the world line is not null, whereupon the derivatives of theHeaviside functions vanish. A second contribution to mass renormalization comes from the ZPF in QED and stochastic electro-dynamics. Notice in this derivation the absence of a problem associated with a factor of 4/3 destroying Lorentzcovariance - see [13] for a nice review of the history of that problem. ichael Ibison A New Case for Direct Actionichael Ibison A New Case for Direct Action Figure 1: Regularized Maxwell self-energy as function of speed. All magnitudes → ∞ as ε → w ( σ ) = σ , so the totalaction is I = − π X σ = ± X p,q e p e q w ( σ ) Z dκ Z dλ ˙ x p ( κ ) ◦ ˙ x q ( λ ) δ (cid:16) ( x p ( κ ) − x q ( λ )) − σε (cid:17) − X q m q Z q dx q , (9)the self action is L self = − lim ε → e π | ε | (cid:16) Θ (cid:0) − v (cid:1) p − v + Θ (cid:0) v − (cid:1) p v − (cid:17) (10)and the self-energy H self = v . ∂L self ∂ v − L self = lim ε → e π | ε | " Θ (cid:0) − v (cid:1) √ − v − Θ (cid:0) v − (cid:1) √ v − (11)is plotted in figure 2. This implies the energy is zero at light speed and is infinitein magnitude at all other speeds (positive infinite at sub-luminal speeds, negative infi-nite at superluminal speeds). The steps leading to (8) are not valid precisely at lightspeed however. An interpretation consistent with (10) is that the charge is comprised5 ichael Ibison A New Case for Direct Actionichael Ibison A New Case for Direct Action Figure 2: Regularized self-energy of charge due to modified direct action. All non-zeromagnitudes → ∞ as ε → | v | = 1 + , − , say).Another alternative reported previously [18] and having the same end result in thiscase is to modify the action in (4) so that the scalar product ˙ x p ( κ ) ◦ ˙ x q ( λ ) is replacedby the positive definite | ˙ x p ( κ ) ◦ ˙ x q ( λ ) | - which change cannot affect the predictions ofthe action in the sub-luminal domain. Then the regularized version of (4) can be written I = − π X σ = ± X p,q e p e q w ( σ ) Z dκ Z dλ | ˙ x p ( κ ) ◦ ˙ x q ( λ ) | δ (cid:16) ( x p ( κ ) − x q ( λ )) − σε (cid:17) − X q m q Z q dx q = − π X σ = ± X p,q e p e q w ( σ ) Z dκ Z dλ δ ( x p ( κ ) − x q ( λ )) − σε ˙ x p ( κ ) ◦ ˙ x q ( λ ) ! − X q m q Z q dx q . (12) The two approaches are not generally equivalent, specifically in the entirely superluminal domain awayfrom the nearly null motion considered here. ichael Ibison A New Case for Direct Actionichael Ibison A New Case for Direct Action
An early attraction of direct action was that it permitted the easy elimination of selfenergy by excluding self-action from the action Schwarzschild [14], Tetrode [15], andFokker [16]. In (4) P p,q → P p,q ; p = q and therefore H self = L self = 0. Later it was re-alized that self-action generates physically observable consequences and so cannot besimply removed - at least from QED. Feynman [17] for example noted that pair creationcan be regarded as a form of promoted self-action. One of the alleged advantages ofdirect action (over field theory) was then lost. Time reversals, which necessitate that at least some part of the trajectory is spacelike, were implicitlyexcluded at the outset by the particular form of the mechanical action. They are not excluded bythe electromagnetic part of the action alone however. Use has been made of the condition that the world line is not null, whereupon the derivatives of theHeaviside functions vanish. A second contribution to mass renormalization comes from the ZPF in QED and stochastic electro-dynamics. Notice in this derivation the absence of a problem associated with a factor of 4/3 destroying Lorentzcovariance - see [13] for a nice review of the history of that problem. ichael Ibison A New Case for Direct Actionichael Ibison A New Case for Direct Action Figure 1: Regularized Maxwell self-energy as function of speed. All magnitudes → ∞ as ε → w ( σ ) = σ , so the totalaction is I = − π X σ = ± X p,q e p e q w ( σ ) Z dκ Z dλ ˙ x p ( κ ) ◦ ˙ x q ( λ ) δ (cid:16) ( x p ( κ ) − x q ( λ )) − σε (cid:17) − X q m q Z q dx q , (9)the self action is L self = − lim ε → e π | ε | (cid:16) Θ (cid:0) − v (cid:1) p − v + Θ (cid:0) v − (cid:1) p v − (cid:17) (10)and the self-energy H self = v . ∂L self ∂ v − L self = lim ε → e π | ε | " Θ (cid:0) − v (cid:1) √ − v − Θ (cid:0) v − (cid:1) √ v − (11)is plotted in figure 2. This implies the energy is zero at light speed and is infinitein magnitude at all other speeds (positive infinite at sub-luminal speeds, negative infi-nite at superluminal speeds). The steps leading to (8) are not valid precisely at lightspeed however. An interpretation consistent with (10) is that the charge is comprised5 ichael Ibison A New Case for Direct Actionichael Ibison A New Case for Direct Action Figure 2: Regularized self-energy of charge due to modified direct action. All non-zeromagnitudes → ∞ as ε → | v | = 1 + , − , say).Another alternative reported previously [18] and having the same end result in thiscase is to modify the action in (4) so that the scalar product ˙ x p ( κ ) ◦ ˙ x q ( λ ) is replacedby the positive definite | ˙ x p ( κ ) ◦ ˙ x q ( λ ) | - which change cannot affect the predictions ofthe action in the sub-luminal domain. Then the regularized version of (4) can be written I = − π X σ = ± X p,q e p e q w ( σ ) Z dκ Z dλ | ˙ x p ( κ ) ◦ ˙ x q ( λ ) | δ (cid:16) ( x p ( κ ) − x q ( λ )) − σε (cid:17) − X q m q Z q dx q = − π X σ = ± X p,q e p e q w ( σ ) Z dκ Z dλ δ ( x p ( κ ) − x q ( λ )) − σε ˙ x p ( κ ) ◦ ˙ x q ( λ ) ! − X q m q Z q dx q . (12) The two approaches are not generally equivalent, specifically in the entirely superluminal domain awayfrom the nearly null motion considered here. ichael Ibison A New Case for Direct Actionichael Ibison A New Case for Direct Action Following the steps in (6) the modified self action is then I self = − e π lim ε → X σ = ± w ( σ ) Z dξ Z dλ δ ( x ( λ + ξ ) − x ( λ )) − σε ˙ x ( λ + ξ ) ◦ ˙ x ( λ ) ! = − lim ε → e π | ε | Z dλ (cid:16) w (1) Θ (cid:0) ˙ x ( λ ) (cid:1) p ˙ x ( λ ) + w ( −
1) Θ (cid:0) − ˙ x ( λ ) (cid:1) p − ˙ x ( λ ) (cid:17) (13)so the effect is just to change the sign of w ( −
1) from how it appears in (7). The modifiedenergy is then the same as (11) provided now w ( σ ) = 1 (figure 2).Since the mechanical mass is no longer required for Coulomb mass renormalization andin any case is infinite at light speed it can now be dropped altogether from the (modified)classical theory, in which case any finite observed mass must here be attributed entirelyto an external interaction in the spirit of Higgs. The simplest possibility is that theexternal mass-giving interaction is entirely electromagnetic necessitating therefore somesort of EM ZPF i.e. at zero Kelvin. As written the self-action (10) is unsatisfactory; the limit cannot be taken in advanceof extremization, and whatever finite non-zero contributions may remain as ε → I self = − Z dλ µ ( λ ) ˙ x ( λ ) − Z d x A ( ext ) ◦ j = − Z dλ ˙ x ( λ ) ◦ (cid:18) µ ( λ ) ˙ x ( λ ) + eA ( ext ) ( x ( λ )) (cid:19) (14)where A ( ext ) is the potential due to non-self sources. µ is an undetermined multipliervariation of which generates the light speed constraint ˙ x ( λ ) = 0. Variation of x ( λ )gives ddλ ( µ ( λ ) ˙ x ( λ )) = eF ( x ( λ )) ◦ ˙ x ( λ ) , (15)where F ab ≡ ∂ a A ( ext ) b − ∂ b A ( ext ) a , ( F ◦ ˙ x ) a = F ab ˙ x b . (16)The mass can be inferred from the dynamics by forming the scalar product of Eq. (15)with ¨ x and using the light speed constraint to give µ = e ¨ x ◦ ( F ◦ ˙ x ) (cid:14) ¨ x .7 ichael Ibison A New Case for Direct Actionichael Ibison A New Case for Direct Action
1) from how it appears in (7). The modifiedenergy is then the same as (11) provided now w ( σ ) = 1 (figure 2).Since the mechanical mass is no longer required for Coulomb mass renormalization andin any case is infinite at light speed it can now be dropped altogether from the (modified)classical theory, in which case any finite observed mass must here be attributed entirelyto an external interaction in the spirit of Higgs. The simplest possibility is that theexternal mass-giving interaction is entirely electromagnetic necessitating therefore somesort of EM ZPF i.e. at zero Kelvin. As written the self-action (10) is unsatisfactory; the limit cannot be taken in advanceof extremization, and whatever finite non-zero contributions may remain as ε → I self = − Z dλ µ ( λ ) ˙ x ( λ ) − Z d x A ( ext ) ◦ j = − Z dλ ˙ x ( λ ) ◦ (cid:18) µ ( λ ) ˙ x ( λ ) + eA ( ext ) ( x ( λ )) (cid:19) (14)where A ( ext ) is the potential due to non-self sources. µ is an undetermined multipliervariation of which generates the light speed constraint ˙ x ( λ ) = 0. Variation of x ( λ )gives ddλ ( µ ( λ ) ˙ x ( λ )) = eF ( x ( λ )) ◦ ˙ x ( λ ) , (15)where F ab ≡ ∂ a A ( ext ) b − ∂ b A ( ext ) a , ( F ◦ ˙ x ) a = F ab ˙ x b . (16)The mass can be inferred from the dynamics by forming the scalar product of Eq. (15)with ¨ x and using the light speed constraint to give µ = e ¨ x ◦ ( F ◦ ˙ x ) (cid:14) ¨ x .7 ichael Ibison A New Case for Direct Actionichael Ibison A New Case for Direct Action Eq. (15) is Lorentz covariant independent of the parameterization and invariantunder simultaneous inversion of mass and charge. It describes only the passive responseof the charge to an existing field however; the active - field-generation - behavior dependson the sign of the charge though not at all on the mass. A description of the 4-currentand momentum that carries sufficient information for both active and passive roles mustdistinguish between the sign of the charge and the sign of the mass. Choosing againlaboratory time λ = x ≡ t then p ≡ | µ | ˙ x defines a null 4-momentum with p > p ∈ R because ( | µ | , ˆ˙ x ) ∈ R + × S ≈ R .A representation that preserves this definition and independently carries the informationabout the sign of the mass and having therefore the dimensionality of ( µ, ˆ˙ x ) ∈ R × S -a double cover of R - can be given in terms of spinors. Direct action provides a useful framework in which to address the problem of infinite self-action of the classical charge. And, of relevance to that problem, it highlights ambiguitiesin, and naturally suggests alternatives to, the application of the Maxwell EM theory tocharges at or near light speed. Two alternatives discussed here suggest a solution tothe problem of infinite self-energy whilst reproducing the behavior of a classical chargein the sub-subluminal domain, i.e. as predicted by direct-action with a component ofmechanical mass action. In the latter case the Maxwell equations remain valid and massrenormalization is required as usual. The modified actions differ in their predictionswhen the mechanical mass action is omitted. In that case, unlike in the traditionalclassical EM theory, the system predicts light-speed motion for a classical point chargewith zero total self-energy in the absence of external fields. With these changes howeverthe correspondence with the Maxwell field theory is destroyed.So far the modified direct-action theories above have not been projected back into anequivalent field theory, though there are features suggestive of a connection with (firstquantized) Dirac theory. Probably the best hope for this kind of approach is conver-gence with a de-Broglie-Bohm style formulation of the single-particle Dirac equation -as promoted for instance by [19] and [20] - wherein the null particle trajectories are theBohmian flow lines. It is sufficient to show that µ transforms like dλ so that dλ/µ is a Lorentz scalar. From Eq. (15),and letting λ = f ( κ ) ⇒ dλ = ˙ fdκ , it follows in a few steps that dλ/µ is independent of f and alsotherefore of however λ may be affected by a transformation. Since EM has no intrinsic length scale the mass-scale must come from external interactions whenintrinsic mechanical mass is excluded from the theory. ichael Ibison A New Case for Direct Actionichael Ibison A New Case for Direct Action
1) from how it appears in (7). The modifiedenergy is then the same as (11) provided now w ( σ ) = 1 (figure 2).Since the mechanical mass is no longer required for Coulomb mass renormalization andin any case is infinite at light speed it can now be dropped altogether from the (modified)classical theory, in which case any finite observed mass must here be attributed entirelyto an external interaction in the spirit of Higgs. The simplest possibility is that theexternal mass-giving interaction is entirely electromagnetic necessitating therefore somesort of EM ZPF i.e. at zero Kelvin. As written the self-action (10) is unsatisfactory; the limit cannot be taken in advanceof extremization, and whatever finite non-zero contributions may remain as ε → I self = − Z dλ µ ( λ ) ˙ x ( λ ) − Z d x A ( ext ) ◦ j = − Z dλ ˙ x ( λ ) ◦ (cid:18) µ ( λ ) ˙ x ( λ ) + eA ( ext ) ( x ( λ )) (cid:19) (14)where A ( ext ) is the potential due to non-self sources. µ is an undetermined multipliervariation of which generates the light speed constraint ˙ x ( λ ) = 0. Variation of x ( λ )gives ddλ ( µ ( λ ) ˙ x ( λ )) = eF ( x ( λ )) ◦ ˙ x ( λ ) , (15)where F ab ≡ ∂ a A ( ext ) b − ∂ b A ( ext ) a , ( F ◦ ˙ x ) a = F ab ˙ x b . (16)The mass can be inferred from the dynamics by forming the scalar product of Eq. (15)with ¨ x and using the light speed constraint to give µ = e ¨ x ◦ ( F ◦ ˙ x ) (cid:14) ¨ x .7 ichael Ibison A New Case for Direct Actionichael Ibison A New Case for Direct Action Eq. (15) is Lorentz covariant independent of the parameterization and invariantunder simultaneous inversion of mass and charge. It describes only the passive responseof the charge to an existing field however; the active - field-generation - behavior dependson the sign of the charge though not at all on the mass. A description of the 4-currentand momentum that carries sufficient information for both active and passive roles mustdistinguish between the sign of the charge and the sign of the mass. Choosing againlaboratory time λ = x ≡ t then p ≡ | µ | ˙ x defines a null 4-momentum with p > p ∈ R because ( | µ | , ˆ˙ x ) ∈ R + × S ≈ R .A representation that preserves this definition and independently carries the informationabout the sign of the mass and having therefore the dimensionality of ( µ, ˆ˙ x ) ∈ R × S -a double cover of R - can be given in terms of spinors. Direct action provides a useful framework in which to address the problem of infinite self-action of the classical charge. And, of relevance to that problem, it highlights ambiguitiesin, and naturally suggests alternatives to, the application of the Maxwell EM theory tocharges at or near light speed. Two alternatives discussed here suggest a solution tothe problem of infinite self-energy whilst reproducing the behavior of a classical chargein the sub-subluminal domain, i.e. as predicted by direct-action with a component ofmechanical mass action. In the latter case the Maxwell equations remain valid and massrenormalization is required as usual. The modified actions differ in their predictionswhen the mechanical mass action is omitted. In that case, unlike in the traditionalclassical EM theory, the system predicts light-speed motion for a classical point chargewith zero total self-energy in the absence of external fields. With these changes howeverthe correspondence with the Maxwell field theory is destroyed.So far the modified direct-action theories above have not been projected back into anequivalent field theory, though there are features suggestive of a connection with (firstquantized) Dirac theory. Probably the best hope for this kind of approach is conver-gence with a de-Broglie-Bohm style formulation of the single-particle Dirac equation -as promoted for instance by [19] and [20] - wherein the null particle trajectories are theBohmian flow lines. It is sufficient to show that µ transforms like dλ so that dλ/µ is a Lorentz scalar. From Eq. (15),and letting λ = f ( κ ) ⇒ dλ = ˙ fdκ , it follows in a few steps that dλ/µ is independent of f and alsotherefore of however λ may be affected by a transformation. Since EM has no intrinsic length scale the mass-scale must come from external interactions whenintrinsic mechanical mass is excluded from the theory. ichael Ibison A New Case for Direct Actionichael Ibison A New Case for Direct Action Acknowledgements
My thanks go to the founder of the PIRT series of conferences Michael Duffy, and toPeter Rowlands for his work to ensure a very enjoyable 2008 meeting at Imperial College.I am grateful to Eric Katerman for helpful discussions on the symmetry group of theaction.
References [1] Wheeler J A and Feynman R P 1945 Interaction with the absorber as the mechanismof radiation Rev. Mod. Phys. 17 157-81[2] Wheeler J A and Feynman R P 1949 Classical electrodynamics in terms of directinterparticle action Rev. Mod. Phys. 21 425-33[3] Davies P C W 1977 The Physics of Time Asymmetry (Berkeley, CA: University ofCalifornia Press)[4] Davies P C W 1972 Is the universe transparent or opaque? J. Phys. A 5 1722-37[5] Price H 1997 Time’s Arrow and Archimedes’ Point: New Directions for the Physicsof TimeOxford University Press, USA)[6] Zeh H D 2007 The physical basis of the direction of time (Berlin: Springer)[7] Ibison M 2006 Are Advanced Potentials Anomalous? in Frontiers of Time: Retro-causation - Experiment and Theory AIP Conference Proceedings ed D P Sheehan pp3-19[8] Davies P C W 1971 Extension of Wheeler-Feynman quantum theory to the relativisticdomain I. Scattering processes J. Phys. A 4 836-45[9] Davies P C W 1972 Extension of Wheeler-Feynman quantum theory to the relativisticdomain II. Emission processes J. Phys. A 5 1024-36[10] Davies P C W 1970 A quantum theory of Wheeler-Feynman electrodynamics Proc.Cam. Phil. Soc. 68 751-64[11] Boyer T H 1980 A brief survey of stochastic electrodynamics Foundations of Radia-tion Theory and Quantum Electrodynamics ed A O Barut (New York: Plenum Press)pp 49-63[12] Itzykson C and Zuber J-B 1985 Quantum field theory (New York: McGraw-Hill)9 ichael Ibison A New Case for Direct Actionichael Ibison A New Case for Direct Action