Renormalized field theory from particle theory
Contrary to conventional wisdom that pretends that particle theory has been disproved and replaced by field theory, we demonstrate that the Hamiltonian particle theory gives an improved version of the usual field-theoretic Hamiltonian, without the difficulties associated to divergences. Moreover, the particle theory is formulated in base to physical charges and masses of particles, whereas field theory only can use unphysical bare quantities.
Newton introduced a model of instantaneous direct interactions among massive particles. This model was latter replicated by Coulomb for charged particles and became known as action-at-a-distance; an unfortunate name has generated unending polemics among physicists and philosophers. A better name is direct-particle-interaction. Physicists, dissatisfied with the action-at-distance model of interactions, introduced a contact-action model based in fields. The mystery on how one particle interacts with other particle without a mediator, \[ \mathrm{particle} \Longleftrightarrow \mathrm{particle} , \] was replaced by a new pair of mysteries: how do particles and fields interact without a mediator? \[ \mathrm{particle} \Longleftrightarrow \mathrm{field} \Longleftrightarrow \mathrm{particle} \] But doubling the number of mysteries was not the only problem. Field theory introduced a broad collection of difficulties; divergences and violations of causality and conservation laws are mentioned often in the literature, but we also have the introduction in the formalism of unobservable systems with an infinite number of degrees of freedom, the introduction of the unphysical bare particles and virtual particles, or the inherent time-asymmetry. General relativity came to only make the situation even worse; General relativity is not an ordinary field theory, because the role of the gravitational field is replaced by a curved spacetime, which introduces further difficulties atop ordinary field theories. Of course, the same physicists that claim that the old action-at-distance model of Newtonian gravity was wmisterious, don't even bother to ask how a massive body curves spacetime or how another body detects the curvature of the spacetime and reacts to it by moving in a different way.
The conventional wisdom is that action-at-a-distance was disproved by experiment in the 19th century, but when one checks the sources for those bold claims one finds that their authors are ignoring the physical and mathematical differences between field theories and General Relativity and the theories of Coulomb and Newton. Consider a standard highly-considered textbook [1]; you can see therein that Steven Weinberg claims on the section 8.3 that the field-theoretic quantity \[ V_\mathrm{field} = \frac{1}{2} \int \mathrm{d}^3 \boldsymbol{x} \int \mathrm{d}^3 \boldsymbol{y} \frac{\rho(\boldsymbol{x})\rho(\boldsymbol{y})}{4\pi\epsilon_0 |\boldsymbol{x}-\boldsymbol{y}|} \] is "the familiar Coulomb energy". His claim is not correct. First, the above expression is fully static, there is no time-dependence, whereas the true Coulomb energy depends on time implicitly via the positions of particles as \( V_\mathrm{Coulomb}(\{\boldsymbol{r}_i(t)\}) \). Second, the expression given by Weinberg is infinite whereas the true Coulomb energy is finite. There are other differences between (3) and the true Coulomb energy \( V_\mathrm{Coulomb} \), but they are more subtle and beyond the scope of this article.
In this article, we will demonstrate how particle theory yields an improved version of field theory. We will restrict the discussion to electromagnetism. We will start with the next energy for an isolated system of \(N\) charges moving at low velocities \[ E = \sum_i \frac{\boldsymbol{p}_i^2}{2m_i} + \frac{1}{2} \sum_i \sum_{j\neq i} \frac{e_ie_j}{4\pi\epsilon_0 | \boldsymbol{r}_i - \boldsymbol{r}_j |} \Big( 1 - \frac{\boldsymbol{p}_i\boldsymbol{p}_j}{m_im_jc^2} \Big) .\] Next we introduce the particle potentials \[ \phi_i \equiv \sum_{j\neq i} \frac{e_j}{4\pi\epsilon_0 | \boldsymbol{r}_i - \boldsymbol{r}_j |} \] \[ \boldsymbol{A}_i \equiv \sum_{j\neq i} \frac{e_j}{4\pi\epsilon_0 | \boldsymbol{r}_i - \boldsymbol{r}_j |} \frac{\boldsymbol{p}_j}{m_jc} ,\] and write a more concise expression for the energy \[ E = \sum_i \Big( \frac{\boldsymbol{p}_i^2}{2m_i} + \frac{1}{2} e_i \phi_i - \frac{1}{2} \frac{e_i\boldsymbol{p}_i}{m_ic} \boldsymbol{A}_i \Big) . \] Field theory is formulated over a \(4D\) spacetime background instead of over a \(6N\) phase space; as a consequence, velocities acquire a more relevant status than momenta. In a first step to derive field theory from particle theory, we need to replace momenta by velocities. Using Hamiltonian equations we can obtain the velocities \[ \boldsymbol{v}_i = \frac{\boldsymbol{p}_i}{m_i} - \sum_{j\neq i} \frac{e_ie_j}{4\pi\epsilon_0 | \boldsymbol{r}_i - \boldsymbol{r}_j |} \frac{\boldsymbol{p}_j}{m_im_jc^2} = \frac{\boldsymbol{p}_i}{m_i} - \frac{e_i}{m_ic} \boldsymbol{A}_i \] to obtain \[ E = \sum_i \Big( \frac{m_i\boldsymbol{v}_i^2}{2} + \frac{1}{2} e_i \phi_i + \frac{1}{2} \frac{e_i\boldsymbol{v}_i}{c} \boldsymbol{A}_i \Big) .\] This expression depends on \(2N\) potentials, whereas classical electrodynamics only deals with a pair of potentials. The reduction is achieved by eliminating the constraint \(j\neq i\). For instance, for the scalar potential, \[ \sum_{j\neq i} \frac{e_j}{4\pi\epsilon_0 | \boldsymbol{r}_i - \boldsymbol{r}_j |} = \sum_j \frac{e_j}{4\pi\epsilon_0 | \boldsymbol{r}_i - \boldsymbol{r}_j |} - \frac{e_i}{4\pi\epsilon_0 | \boldsymbol{r}_i - \boldsymbol{r}_i |} ,\] which produces \[ \phi_i = \phi - \phi_i^\mathrm{self} ,\] with a similar expression for the vector potential. The energy is now \[ E = \sum_i \frac{m_i\boldsymbol{v}_i^2}{2} + \frac{1}{2} \sum_i \Big[ e_i \Big( \phi - \phi_i^\mathrm{self} \Big) + \frac{e_i\boldsymbol{v}_i}{c} \Big( \boldsymbol{A} - \boldsymbol{A}_i^\mathrm{self} \Big) \Big] .\] The new potentials \(\phi\) and \(\boldsymbol{A}\) diverge, but \(\phi_i^\mathrm{self}\) and \(\boldsymbol{A}_i^\mathrm{self}\) eliminate this divergence and the total energy \(E\) continues being a finite quantity. We will see below that the lack of those \(N\) correction terms in field theory is the reason why field theory predicts nonsensical infinite energies.
The above expression can be re-organized as \[ E = \frac{1}{2} \sum_i \Big( m_i\boldsymbol{v}_i^2 - \frac{e_i\boldsymbol{v}_i}{c} \boldsymbol{A}_i^\mathrm{self} \Big) + \frac{1}{2} \sum_i \Big( e_i \phi + \frac{e_i\boldsymbol{v}_i}{c} \boldsymbol{A} \Big) - E^\mathrm{self} ,\] where \( E^\mathrm{self} \equiv \frac{1}{2} \sum_i e_i \phi_i^\mathrm{self} \). Multiplying both sides of (8) with \(m_i\), replacing \(\boldsymbol{A}_i\) by \((\boldsymbol{A} - \boldsymbol{A}_i^\mathrm{self})\) and reorganizing the result gives \[ \Big( m_i \boldsymbol{v}_i - \frac{e_i}{c} \boldsymbol{A}_i^\mathrm{self} \Big) = \boldsymbol{p}_i - \frac{e_i}{c} \boldsymbol{A} .\] Multiplying again both sides by \( \boldsymbol{v}_i\) and iterating finally produces the identity \[ \Big( m_i \boldsymbol{v}_i^2 - \frac{e_i\boldsymbol{v}_i}{c} \boldsymbol{A}_i^\mathrm{self} \Big) = \Big( \boldsymbol{p}_i - \frac{e_i}{c} \boldsymbol{A} \Big) \frac{1}{m_i - m_i^\mathrm{self}} \Big( \boldsymbol{p}_i - \frac{e_i}{c} \boldsymbol{A} \Big) ,\] where \(m_i^\mathrm{self} \equiv (e_i / \boldsymbol{v}_i c) \boldsymbol{A}_i^\mathrm{self} \), which is evidently a divergent quantity, sometimes named the "self-mass" or the "electromagnetic mass" in the field-theoretic literature. Replacing \( \boldsymbol{A}_i^\mathrm{self}\) by its value, expanding it on a power series on \((v/c)\) and retaining only the first term in the series we obtain \[ m_i^\mathrm{self} = \frac{e_i^2}{4\pi\epsilon_0 |\boldsymbol{r}_i -\boldsymbol{r}_i| c^2} ,\] a result first obtained by Dirac. The above identity (15) can be used on (13) to give \[ E = \sum_i \frac{1}{2} \frac{1}{m_i - m_i^\mathrm{self}} \Big( \boldsymbol{p}_i - \frac{e_i}{c} \boldsymbol{A} \Big)^2 + \frac{1}{2} \sum_i \Big( e_i \phi + \frac{e_i\boldsymbol{v}_i}{c} \boldsymbol{A} \Big) - E^\mathrm{self} \] Finally using densities \(\rho(\boldsymbol{z},t) \equiv \sum_i e_i \delta(\boldsymbol{z} - \boldsymbol{r}_i(t))\) and currents \(\boldsymbol{j}(\boldsymbol{z},t) \equiv \sum_i e_i \boldsymbol{v}_i\delta(\boldsymbol{z} - \boldsymbol{r}_i(t))\) and combining both masses on a new concept of mass as \(m_i^\mathrm{bare} \equiv m_i - m_i^\mathrm{self}\) yields \[ E = \sum_i \frac{1}{2m_i^\mathrm{bare}} \Big( \boldsymbol{p}_i - \frac{e_i}{c} \boldsymbol{A} \Big)^2 + \frac{1}{2} \int \mathrm{d}V \Big( \rho \phi + \boldsymbol{j} \boldsymbol{A} \Big) - E^\mathrm{self} .\] The integral can be written in an alternative form, obtaining the final result as function of electric \(\boldsymbol{E}\) and magnetic fields \(\boldsymbol{B}\) \[ E = \sum_i \frac{1}{2m_i^\mathrm{bare}} \Big( \boldsymbol{p}_i - \frac{e_i}{c} \boldsymbol{A} \Big)^2 + \frac{1}{8\pi} \int \mathrm{d}V \Big( \boldsymbol{E}^2 + \boldsymbol{B}^2 \Big) - E^\mathrm{self} .\] Except by the correction term \(E^\mathrm{self}\), this expression is fully analogous to the one proposed by field theory. The difference being in the physical interpretation; the integral does not represent a separate system, the field, with its own degrees of freedoms; but the integral simply represents a component of the full interaction between a system of particles.
We also can split the electric field into longitudinal and transversal components \( \boldsymbol{E}^2 = \boldsymbol{E}_\mathrm{L}^2 + \boldsymbol{E}_\mathrm{T}^2\) and then use the relation \[ \frac{1}{2} \int \boldsymbol{E}_\mathrm{L}^2 \mathrm{d}V = \frac{1}{2} \sum_i\sum_{j\neq i} \frac{e_i e_j}{4\pi\epsilon_0 | \boldsymbol{r}_i - \boldsymbol{r}_j |} + E^\mathrm{self} \] to write the energy in the alternative form \[ E = \sum_i \frac{1}{2m_i^\mathrm{bare}} \Big( \boldsymbol{p}_i - \frac{e_i}{c} \boldsymbol{A} \Big)^2 + \frac{1}{2} \sum_i\sum_{j\neq i} \frac{e_i e_j}{4\pi\epsilon_0 | \boldsymbol{r}_i - \boldsymbol{r}_j |} + \frac{1}{8\pi} \int \mathrm{d}V \Big( \boldsymbol{E}_\mathrm{T}^2 + \boldsymbol{B}^2 \Big) .\] Note that field theory is lacking the \(E^\mathrm{self}\) term in (19); therefore physicists only can obtain (21) when using the next nonsensical expression \[ \frac{1}{2} \int \boldsymbol{E}_\mathrm{L}^2 \mathrm{d}V = \frac{1}{2} \sum_i\sum_j \frac{e_i e_j}{4\pi\epsilon_0 | \boldsymbol{r}_i - \boldsymbol{r}_j |} = \frac{1}{2} \sum_i\sum_{j\neq i} \frac{e_i e_j}{4\pi\epsilon_0 | \boldsymbol{r}_i - \boldsymbol{r}_j |} \] This has the same validity than writing \((\infty = 6)\). Despite being nonsensical, the expression (22) is found in the mainstream literature; it is equation 1.57 in textbook [2].
We have demonstrated that particle theory is superior to field theory, and that field theoretic expressions such as (3) cannot even reproduce ancient Coulomb energy; not at least without using certain amounts of mathematical funambulism, such as making \((\infty-\infty)\) equal to the value required by experiment or pretending that \(\Phi(R(t)) = \Phi(\boldsymbol{r},t)\).
[1] The Quantum Theory of Fields, Volume I 1996: Cambridge University Press; Cambridge. Weinberg, Steven.
[2] Quantum Field Theory, Revised Edition 1999: John Wiley & Sons Ltd.; Chichester. Mandl, F.; Shaw, G.
Newton introduced a model of instantaneous direct interactions among massive particles. This model was latter replicated by Coulomb for charged particles and became known as action-at-a-distance; an unfortunate name has generated unending polemics among physicists and philosophers. A better name is direct-particle-interaction. Physicists, dissatisfied with the action-at-distance model of interactions, introduced a contact-action model based in fields. The mystery on how one particle interacts with other particle without a mediator, \[ \mathrm{particle} \Longleftrightarrow \mathrm{particle} , \] was replaced by a new pair of mysteries: how do particles and fields interact without a mediator? \[ \mathrm{particle} \Longleftrightarrow \mathrm{field} \Longleftrightarrow \mathrm{particle} \] But doubling the number of mysteries was not the only problem. Field theory introduced a broad collection of difficulties; divergences and violations of causality and conservation laws are mentioned often in the literature, but we also have the introduction in the formalism of unobservable systems with an infinite number of degrees of freedom, the introduction of the unphysical bare particles and virtual particles, or the inherent time-asymmetry. General relativity came to only make the situation even worse; General relativity is not an ordinary field theory, because the role of the gravitational field is replaced by a curved spacetime, which introduces further difficulties atop ordinary field theories. Of course, the same physicists that claim that the old action-at-distance model of Newtonian gravity was wmisterious, don't even bother to ask how a massive body curves spacetime or how another body detects the curvature of the spacetime and reacts to it by moving in a different way.
The conventional wisdom is that action-at-a-distance was disproved by experiment in the 19th century, but when one checks the sources for those bold claims one finds that their authors are ignoring the physical and mathematical differences between field theories and General Relativity and the theories of Coulomb and Newton. Consider a standard highly-considered textbook [1]; you can see therein that Steven Weinberg claims on the section 8.3 that the field-theoretic quantity \[ V_\mathrm{field} = \frac{1}{2} \int \mathrm{d}^3 \boldsymbol{x} \int \mathrm{d}^3 \boldsymbol{y} \frac{\rho(\boldsymbol{x})\rho(\boldsymbol{y})}{4\pi\epsilon_0 |\boldsymbol{x}-\boldsymbol{y}|} \] is "the familiar Coulomb energy". His claim is not correct. First, the above expression is fully static, there is no time-dependence, whereas the true Coulomb energy depends on time implicitly via the positions of particles as \( V_\mathrm{Coulomb}(\{\boldsymbol{r}_i(t)\}) \). Second, the expression given by Weinberg is infinite whereas the true Coulomb energy is finite. There are other differences between (3) and the true Coulomb energy \( V_\mathrm{Coulomb} \), but they are more subtle and beyond the scope of this article.
In this article, we will demonstrate how particle theory yields an improved version of field theory. We will restrict the discussion to electromagnetism. We will start with the next energy for an isolated system of \(N\) charges moving at low velocities \[ E = \sum_i \frac{\boldsymbol{p}_i^2}{2m_i} + \frac{1}{2} \sum_i \sum_{j\neq i} \frac{e_ie_j}{4\pi\epsilon_0 | \boldsymbol{r}_i - \boldsymbol{r}_j |} \Big( 1 - \frac{\boldsymbol{p}_i\boldsymbol{p}_j}{m_im_jc^2} \Big) .\] Next we introduce the particle potentials \[ \phi_i \equiv \sum_{j\neq i} \frac{e_j}{4\pi\epsilon_0 | \boldsymbol{r}_i - \boldsymbol{r}_j |} \] \[ \boldsymbol{A}_i \equiv \sum_{j\neq i} \frac{e_j}{4\pi\epsilon_0 | \boldsymbol{r}_i - \boldsymbol{r}_j |} \frac{\boldsymbol{p}_j}{m_jc} ,\] and write a more concise expression for the energy \[ E = \sum_i \Big( \frac{\boldsymbol{p}_i^2}{2m_i} + \frac{1}{2} e_i \phi_i - \frac{1}{2} \frac{e_i\boldsymbol{p}_i}{m_ic} \boldsymbol{A}_i \Big) . \] Field theory is formulated over a \(4D\) spacetime background instead of over a \(6N\) phase space; as a consequence, velocities acquire a more relevant status than momenta. In a first step to derive field theory from particle theory, we need to replace momenta by velocities. Using Hamiltonian equations we can obtain the velocities \[ \boldsymbol{v}_i = \frac{\boldsymbol{p}_i}{m_i} - \sum_{j\neq i} \frac{e_ie_j}{4\pi\epsilon_0 | \boldsymbol{r}_i - \boldsymbol{r}_j |} \frac{\boldsymbol{p}_j}{m_im_jc^2} = \frac{\boldsymbol{p}_i}{m_i} - \frac{e_i}{m_ic} \boldsymbol{A}_i \] to obtain \[ E = \sum_i \Big( \frac{m_i\boldsymbol{v}_i^2}{2} + \frac{1}{2} e_i \phi_i + \frac{1}{2} \frac{e_i\boldsymbol{v}_i}{c} \boldsymbol{A}_i \Big) .\] This expression depends on \(2N\) potentials, whereas classical electrodynamics only deals with a pair of potentials. The reduction is achieved by eliminating the constraint \(j\neq i\). For instance, for the scalar potential, \[ \sum_{j\neq i} \frac{e_j}{4\pi\epsilon_0 | \boldsymbol{r}_i - \boldsymbol{r}_j |} = \sum_j \frac{e_j}{4\pi\epsilon_0 | \boldsymbol{r}_i - \boldsymbol{r}_j |} - \frac{e_i}{4\pi\epsilon_0 | \boldsymbol{r}_i - \boldsymbol{r}_i |} ,\] which produces \[ \phi_i = \phi - \phi_i^\mathrm{self} ,\] with a similar expression for the vector potential. The energy is now \[ E = \sum_i \frac{m_i\boldsymbol{v}_i^2}{2} + \frac{1}{2} \sum_i \Big[ e_i \Big( \phi - \phi_i^\mathrm{self} \Big) + \frac{e_i\boldsymbol{v}_i}{c} \Big( \boldsymbol{A} - \boldsymbol{A}_i^\mathrm{self} \Big) \Big] .\] The new potentials \(\phi\) and \(\boldsymbol{A}\) diverge, but \(\phi_i^\mathrm{self}\) and \(\boldsymbol{A}_i^\mathrm{self}\) eliminate this divergence and the total energy \(E\) continues being a finite quantity. We will see below that the lack of those \(N\) correction terms in field theory is the reason why field theory predicts nonsensical infinite energies.
The above expression can be re-organized as \[ E = \frac{1}{2} \sum_i \Big( m_i\boldsymbol{v}_i^2 - \frac{e_i\boldsymbol{v}_i}{c} \boldsymbol{A}_i^\mathrm{self} \Big) + \frac{1}{2} \sum_i \Big( e_i \phi + \frac{e_i\boldsymbol{v}_i}{c} \boldsymbol{A} \Big) - E^\mathrm{self} ,\] where \( E^\mathrm{self} \equiv \frac{1}{2} \sum_i e_i \phi_i^\mathrm{self} \). Multiplying both sides of (8) with \(m_i\), replacing \(\boldsymbol{A}_i\) by \((\boldsymbol{A} - \boldsymbol{A}_i^\mathrm{self})\) and reorganizing the result gives \[ \Big( m_i \boldsymbol{v}_i - \frac{e_i}{c} \boldsymbol{A}_i^\mathrm{self} \Big) = \boldsymbol{p}_i - \frac{e_i}{c} \boldsymbol{A} .\] Multiplying again both sides by \( \boldsymbol{v}_i\) and iterating finally produces the identity \[ \Big( m_i \boldsymbol{v}_i^2 - \frac{e_i\boldsymbol{v}_i}{c} \boldsymbol{A}_i^\mathrm{self} \Big) = \Big( \boldsymbol{p}_i - \frac{e_i}{c} \boldsymbol{A} \Big) \frac{1}{m_i - m_i^\mathrm{self}} \Big( \boldsymbol{p}_i - \frac{e_i}{c} \boldsymbol{A} \Big) ,\] where \(m_i^\mathrm{self} \equiv (e_i / \boldsymbol{v}_i c) \boldsymbol{A}_i^\mathrm{self} \), which is evidently a divergent quantity, sometimes named the "self-mass" or the "electromagnetic mass" in the field-theoretic literature. Replacing \( \boldsymbol{A}_i^\mathrm{self}\) by its value, expanding it on a power series on \((v/c)\) and retaining only the first term in the series we obtain \[ m_i^\mathrm{self} = \frac{e_i^2}{4\pi\epsilon_0 |\boldsymbol{r}_i -\boldsymbol{r}_i| c^2} ,\] a result first obtained by Dirac. The above identity (15) can be used on (13) to give \[ E = \sum_i \frac{1}{2} \frac{1}{m_i - m_i^\mathrm{self}} \Big( \boldsymbol{p}_i - \frac{e_i}{c} \boldsymbol{A} \Big)^2 + \frac{1}{2} \sum_i \Big( e_i \phi + \frac{e_i\boldsymbol{v}_i}{c} \boldsymbol{A} \Big) - E^\mathrm{self} \] Finally using densities \(\rho(\boldsymbol{z},t) \equiv \sum_i e_i \delta(\boldsymbol{z} - \boldsymbol{r}_i(t))\) and currents \(\boldsymbol{j}(\boldsymbol{z},t) \equiv \sum_i e_i \boldsymbol{v}_i\delta(\boldsymbol{z} - \boldsymbol{r}_i(t))\) and combining both masses on a new concept of mass as \(m_i^\mathrm{bare} \equiv m_i - m_i^\mathrm{self}\) yields \[ E = \sum_i \frac{1}{2m_i^\mathrm{bare}} \Big( \boldsymbol{p}_i - \frac{e_i}{c} \boldsymbol{A} \Big)^2 + \frac{1}{2} \int \mathrm{d}V \Big( \rho \phi + \boldsymbol{j} \boldsymbol{A} \Big) - E^\mathrm{self} .\] The integral can be written in an alternative form, obtaining the final result as function of electric \(\boldsymbol{E}\) and magnetic fields \(\boldsymbol{B}\) \[ E = \sum_i \frac{1}{2m_i^\mathrm{bare}} \Big( \boldsymbol{p}_i - \frac{e_i}{c} \boldsymbol{A} \Big)^2 + \frac{1}{8\pi} \int \mathrm{d}V \Big( \boldsymbol{E}^2 + \boldsymbol{B}^2 \Big) - E^\mathrm{self} .\] Except by the correction term \(E^\mathrm{self}\), this expression is fully analogous to the one proposed by field theory. The difference being in the physical interpretation; the integral does not represent a separate system, the field, with its own degrees of freedoms; but the integral simply represents a component of the full interaction between a system of particles.
We also can split the electric field into longitudinal and transversal components \( \boldsymbol{E}^2 = \boldsymbol{E}_\mathrm{L}^2 + \boldsymbol{E}_\mathrm{T}^2\) and then use the relation \[ \frac{1}{2} \int \boldsymbol{E}_\mathrm{L}^2 \mathrm{d}V = \frac{1}{2} \sum_i\sum_{j\neq i} \frac{e_i e_j}{4\pi\epsilon_0 | \boldsymbol{r}_i - \boldsymbol{r}_j |} + E^\mathrm{self} \] to write the energy in the alternative form \[ E = \sum_i \frac{1}{2m_i^\mathrm{bare}} \Big( \boldsymbol{p}_i - \frac{e_i}{c} \boldsymbol{A} \Big)^2 + \frac{1}{2} \sum_i\sum_{j\neq i} \frac{e_i e_j}{4\pi\epsilon_0 | \boldsymbol{r}_i - \boldsymbol{r}_j |} + \frac{1}{8\pi} \int \mathrm{d}V \Big( \boldsymbol{E}_\mathrm{T}^2 + \boldsymbol{B}^2 \Big) .\] Note that field theory is lacking the \(E^\mathrm{self}\) term in (19); therefore physicists only can obtain (21) when using the next nonsensical expression \[ \frac{1}{2} \int \boldsymbol{E}_\mathrm{L}^2 \mathrm{d}V = \frac{1}{2} \sum_i\sum_j \frac{e_i e_j}{4\pi\epsilon_0 | \boldsymbol{r}_i - \boldsymbol{r}_j |} = \frac{1}{2} \sum_i\sum_{j\neq i} \frac{e_i e_j}{4\pi\epsilon_0 | \boldsymbol{r}_i - \boldsymbol{r}_j |} \] This has the same validity than writing \((\infty = 6)\). Despite being nonsensical, the expression (22) is found in the mainstream literature; it is equation 1.57 in textbook [2].
We have demonstrated that particle theory is superior to field theory, and that field theoretic expressions such as (3) cannot even reproduce ancient Coulomb energy; not at least without using certain amounts of mathematical funambulism, such as making \((\infty-\infty)\) equal to the value required by experiment or pretending that \(\Phi(R(t)) = \Phi(\boldsymbol{r},t)\).
References and Notes
[1] The Quantum Theory of Fields, Volume I 1996: Cambridge University Press; Cambridge. Weinberg, Steven.
[2] Quantum Field Theory, Revised Edition 1999: John Wiley & Sons Ltd.; Chichester. Mandl, F.; Shaw, G.