Renormalization as a direct-particle-action correction to field theory
One of the notorious difficulties with field theory is its prediction of nonsensical infinite values for physical properties such as energy. Renormalization is a procedure by which divergent parts of a calculation, leading to the nonsensical infinite results, are absorbed by redefinition into a few measurable quantities, so yielding finite answers. This work shows that renormalization counterterms added to field-theoretic Hamiltonians and Lagrangians are a consequence of direct-particle-action corrections to field theory. Some widespread misunderstandings are also corrected.
Most physicists ignore the physical and mathematical differences between direct-particle-actions on Coulomb and Newton theories and contact-actions in field theories and General Relativity. If you check standard textbooks, as the one by Steven Weinberg [1], you can see that he 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 static, whereas the true Coulomb energy depends on time implicitly via particle positions as \( V_\mathrm{Coulomb}(\boldsymbol{r}_1(t),\boldsymbol{r}_2(t)) \). Second, the expression given by Weinberg is infinite whereas the true Coulomb energy is finite. There are other differences, but they are more subtle and not of interest for the goals of this article.
If you check the revision of classical electrodynamics given in the section 1.4.1 of the textbook by Mandl and Shaw [2], you can see that, on contrast with Weinberg, the pair of authors allow the integral to carry out an explicit time dependence \[ \frac{1}{2} \int \boldsymbol{E}_\mathrm{L}^2 \mathrm{d}^3 \boldsymbol{x} = \frac{1}{2} \int \mathrm{d}^3 \boldsymbol{x} \int \mathrm{d}^3 \boldsymbol{y} \frac{\rho(\boldsymbol{x},t)\rho(\boldsymbol{y},t)}{4\pi\epsilon_0 |\boldsymbol{x}-\boldsymbol{y}|} . \] Introducing the usual charge density \( \rho(\boldsymbol{z},t) = \sum_i e_i \delta(\boldsymbol{z} - \boldsymbol{r}_i(t)) \), Mandl and Shaw write \[ \frac{1}{2} \int \boldsymbol{E}_\mathrm{L}^2 \mathrm{d}^3 \boldsymbol{x} = \frac{1}{2} \sum_i\sum_{j} \frac{e_i e_j}{4\pi\epsilon_0 |\boldsymbol{r}_i -\boldsymbol{r}_j|} = \frac{1}{2} \sum_i\sum_{j\neq i} \frac{e_i e_j}{4\pi\epsilon_0 |\boldsymbol{r}_i -\boldsymbol{r}_j|} ,\] with they stating that the last expression, the one where they have "dropped the infinite self-energy which occurs for point charges" is "the Coulomb interaction". Ignoring subtle issues arising from the difference between time-implicit and time-explicit dependences [3], their equation is so meaningless as writing \( (\infty = 7) \).
By simplicity we will work in the classical and nonrelativistic regime. The energy with direct-particle interactions is given in this limit by \[ E = \sum_i \frac{\boldsymbol{p}_i^2}{2 m_i} + \frac{1}{2} \sum_i\sum_{j\neq i} \frac{e_i e_j}{4\pi\epsilon_0 |\boldsymbol{r}_i -\boldsymbol{r}_j|} \left( 1 - \frac{\boldsymbol{p}_i \boldsymbol{p}_j}{m_i m_j c^2} \right) ,\] which is a finite quantity, because self-interactions are dropped by \( j\neq i \). With the help of a Kronecker delta we can rewrite the summation in a more symmetrical way \[ E = \sum_i \frac{\boldsymbol{p}_i^2}{2 m_i} + \frac{1}{2} \sum_i\sum_j \frac{e_i e_j}{4\pi\epsilon_0 |\boldsymbol{r}_i -\boldsymbol{r}_j|} \left( 1 - \frac{\boldsymbol{p}_i \boldsymbol{p}_j}{m_i m_j c^2} \right) \left( 1 - \delta_{ij} \right) .\] Of course, this continues being a finite quantity. Partially reorganizing the expression yields \[ E = \frac{1}{2} \sum_i \left( \frac{\boldsymbol{p}_i^2}{m_i} + \frac{e_i^2}{4\pi\epsilon_0 |\boldsymbol{r}_i -\boldsymbol{r}_i|} \frac{\boldsymbol{p}_i^2}{m_i^2 c^2} \right) + \frac{1}{2} \sum_i\sum_j \frac{e_i e_j}{4\pi\epsilon_0 |\boldsymbol{r}_i -\boldsymbol{r}_j|} \left( 1 - \frac{\boldsymbol{p}_i \boldsymbol{p}_j}{m_i m_j c^2} \right) + V_\mathrm{countertem} ,\] with the divergent counterterm defined by \[ V_\mathrm{countertem} = - \frac{1}{2} \sum_i \frac{e_i^2}{4\pi\epsilon_0 |\boldsymbol{r}_i -\boldsymbol{r}_i|} .\] Finally using densities and currents we obtain \[ E = \frac{1}{2} \sum_i \left( \frac{\boldsymbol{p}_i^2}{m_i} + \frac{e_i^2}{4\pi\epsilon_0 |\boldsymbol{r}_i -\boldsymbol{r}_i|} \frac{\boldsymbol{p}_i^2}{m_i^2 c^2} \right) + \frac{1}{2} \int \mathrm{d}^3 \boldsymbol{x} \int \mathrm{d}^3 \boldsymbol{y} \frac{\rho(\boldsymbol{x},t)\rho(\boldsymbol{y},t)}{4\pi\epsilon_0 |\boldsymbol{x}-\boldsymbol{y}|} - \int \mathrm{d}^3 \boldsymbol{x} \; \boldsymbol{j}(\boldsymbol{x},t)\boldsymbol{A}(\boldsymbol{x},t) + V_\mathrm{countertem} .\] Although some individual terms are divergent, this overall expression for the energy continues being finite and equivalent to the starting expression (4). Field theory ignores the counterterm, introduces a divergent self-mass concept first obtained by Dirac \[ m_i^\mathrm{self} = \frac{e_i^2}{4\pi\epsilon_0 |\boldsymbol{r}_i -\boldsymbol{r}_i| c^2} \] and reabsorbs the difference between the real mass \( m_i \) and the self-mass by introducing still another concept of divergent mass, the bare mass \( m_i^\mathrm{bare} \), to get the field-theoretic expression \[ E_\mathrm{field} = \frac{1}{2} \sum_i \frac{\boldsymbol{p}_i^\mathrm{bare}\boldsymbol{p}_i^\mathrm{bare}}{m_i^\mathrm{bare}} + \frac{1}{2} \int \mathrm{d}^3 \boldsymbol{x} \int \mathrm{d}^3 \boldsymbol{y} \frac{\rho(\boldsymbol{x},t)\rho(\boldsymbol{y},t)}{4\pi\epsilon_0 |\boldsymbol{x}-\boldsymbol{y}|} - \int \mathrm{d}^3 \boldsymbol{x} \; \boldsymbol{j}(\boldsymbol{x},t)\boldsymbol{A}(\boldsymbol{x},t) ,\] which is not only infinite, but uses the unphysical concept of bare mass. Comparing the expressions (4) and (10), we can easily obtain the fundamental relation between the true Coulomb energy and the field-theoretic expression —for many applications \( \rho(\boldsymbol{z};t) \) [3] can be safely replaced by \( \rho(\boldsymbol{z},t) \) because the functional difference is harmless— \[ \frac{1}{2} \sum_i\sum_{j\neq i} \frac{e_i e_j}{4\pi\epsilon_0 |\boldsymbol{r}_i -\boldsymbol{r}_j|} = \frac{1}{2} \int \mathrm{d}^3 \boldsymbol{x} \int \mathrm{d}^3 \boldsymbol{y} \frac{\rho(\boldsymbol{x},t)\rho(\boldsymbol{y},t)}{4\pi\epsilon_0 |\boldsymbol{x}-\boldsymbol{y}|} - \frac{1}{2} \sum_i \frac{e_i^2}{4\pi\epsilon_0 |\boldsymbol{r}_i -\boldsymbol{r}_i|} \] or, using a more concise notation, \[ V_\mathrm{Coulomb} = V_\mathrm{field} + V_\mathrm{countertem} .\] Of course the infinite counterterm exactly cancels the divergence within the field-theoretic expression giving a finite Coulomb energy in agreement with experiments. We have derived from the direct-particle interaction the field-theoretic interaction plus a renormalization counterterm. The existence of this counterterm is a physical consequence of the fact that the interaction between charged particles cannot be fully described in terms of interacting fields \( \rho(\boldsymbol{x},t)\phi(\boldsymbol{x},t) \). The origin of this impossibility can be traced back to the Coulomb interaction describing the physics of the correlation between particles, and this correlation is beyond a product of spacetime densities \( \rho(\boldsymbol{x},t)\rho(\boldsymbol{y},t) \). Precisely, the absence of correlations in the field-theoretic approach is the ultimate reason why a physical field is required to carry out the interaction between particles, whereas no field is needed in direct-particle-action theories.
The same line or reasoning can be applied in a quantum context, except that then one obtains also a renormalization of electron charge and a final set of quantum electrodynamic expressions in terms of bare masses and bare charges, both unphysical and different from real masses and charges. Using direct-particle gravitational interactions we can obtain general relativity expression plus renormalization corrections.
[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.
[3] Sometimes the density is better written like \( \rho(\boldsymbol{z};t) \) with the semicolon indicating a time-implicit dependence through the particles positions \( \rho(\boldsymbol{z};t) = \rho(\boldsymbol{z},\{\boldsymbol{r}_i(t)\}) \).
Most physicists ignore the physical and mathematical differences between direct-particle-actions on Coulomb and Newton theories and contact-actions in field theories and General Relativity. If you check standard textbooks, as the one by Steven Weinberg [1], you can see that he 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 static, whereas the true Coulomb energy depends on time implicitly via particle positions as \( V_\mathrm{Coulomb}(\boldsymbol{r}_1(t),\boldsymbol{r}_2(t)) \). Second, the expression given by Weinberg is infinite whereas the true Coulomb energy is finite. There are other differences, but they are more subtle and not of interest for the goals of this article.
If you check the revision of classical electrodynamics given in the section 1.4.1 of the textbook by Mandl and Shaw [2], you can see that, on contrast with Weinberg, the pair of authors allow the integral to carry out an explicit time dependence \[ \frac{1}{2} \int \boldsymbol{E}_\mathrm{L}^2 \mathrm{d}^3 \boldsymbol{x} = \frac{1}{2} \int \mathrm{d}^3 \boldsymbol{x} \int \mathrm{d}^3 \boldsymbol{y} \frac{\rho(\boldsymbol{x},t)\rho(\boldsymbol{y},t)}{4\pi\epsilon_0 |\boldsymbol{x}-\boldsymbol{y}|} . \] Introducing the usual charge density \( \rho(\boldsymbol{z},t) = \sum_i e_i \delta(\boldsymbol{z} - \boldsymbol{r}_i(t)) \), Mandl and Shaw write \[ \frac{1}{2} \int \boldsymbol{E}_\mathrm{L}^2 \mathrm{d}^3 \boldsymbol{x} = \frac{1}{2} \sum_i\sum_{j} \frac{e_i e_j}{4\pi\epsilon_0 |\boldsymbol{r}_i -\boldsymbol{r}_j|} = \frac{1}{2} \sum_i\sum_{j\neq i} \frac{e_i e_j}{4\pi\epsilon_0 |\boldsymbol{r}_i -\boldsymbol{r}_j|} ,\] with they stating that the last expression, the one where they have "dropped the infinite self-energy which occurs for point charges" is "the Coulomb interaction". Ignoring subtle issues arising from the difference between time-implicit and time-explicit dependences [3], their equation is so meaningless as writing \( (\infty = 7) \).
By simplicity we will work in the classical and nonrelativistic regime. The energy with direct-particle interactions is given in this limit by \[ E = \sum_i \frac{\boldsymbol{p}_i^2}{2 m_i} + \frac{1}{2} \sum_i\sum_{j\neq i} \frac{e_i e_j}{4\pi\epsilon_0 |\boldsymbol{r}_i -\boldsymbol{r}_j|} \left( 1 - \frac{\boldsymbol{p}_i \boldsymbol{p}_j}{m_i m_j c^2} \right) ,\] which is a finite quantity, because self-interactions are dropped by \( j\neq i \). With the help of a Kronecker delta we can rewrite the summation in a more symmetrical way \[ E = \sum_i \frac{\boldsymbol{p}_i^2}{2 m_i} + \frac{1}{2} \sum_i\sum_j \frac{e_i e_j}{4\pi\epsilon_0 |\boldsymbol{r}_i -\boldsymbol{r}_j|} \left( 1 - \frac{\boldsymbol{p}_i \boldsymbol{p}_j}{m_i m_j c^2} \right) \left( 1 - \delta_{ij} \right) .\] Of course, this continues being a finite quantity. Partially reorganizing the expression yields \[ E = \frac{1}{2} \sum_i \left( \frac{\boldsymbol{p}_i^2}{m_i} + \frac{e_i^2}{4\pi\epsilon_0 |\boldsymbol{r}_i -\boldsymbol{r}_i|} \frac{\boldsymbol{p}_i^2}{m_i^2 c^2} \right) + \frac{1}{2} \sum_i\sum_j \frac{e_i e_j}{4\pi\epsilon_0 |\boldsymbol{r}_i -\boldsymbol{r}_j|} \left( 1 - \frac{\boldsymbol{p}_i \boldsymbol{p}_j}{m_i m_j c^2} \right) + V_\mathrm{countertem} ,\] with the divergent counterterm defined by \[ V_\mathrm{countertem} = - \frac{1}{2} \sum_i \frac{e_i^2}{4\pi\epsilon_0 |\boldsymbol{r}_i -\boldsymbol{r}_i|} .\] Finally using densities and currents we obtain \[ E = \frac{1}{2} \sum_i \left( \frac{\boldsymbol{p}_i^2}{m_i} + \frac{e_i^2}{4\pi\epsilon_0 |\boldsymbol{r}_i -\boldsymbol{r}_i|} \frac{\boldsymbol{p}_i^2}{m_i^2 c^2} \right) + \frac{1}{2} \int \mathrm{d}^3 \boldsymbol{x} \int \mathrm{d}^3 \boldsymbol{y} \frac{\rho(\boldsymbol{x},t)\rho(\boldsymbol{y},t)}{4\pi\epsilon_0 |\boldsymbol{x}-\boldsymbol{y}|} - \int \mathrm{d}^3 \boldsymbol{x} \; \boldsymbol{j}(\boldsymbol{x},t)\boldsymbol{A}(\boldsymbol{x},t) + V_\mathrm{countertem} .\] Although some individual terms are divergent, this overall expression for the energy continues being finite and equivalent to the starting expression (4). Field theory ignores the counterterm, introduces a divergent self-mass concept first obtained by Dirac \[ m_i^\mathrm{self} = \frac{e_i^2}{4\pi\epsilon_0 |\boldsymbol{r}_i -\boldsymbol{r}_i| c^2} \] and reabsorbs the difference between the real mass \( m_i \) and the self-mass by introducing still another concept of divergent mass, the bare mass \( m_i^\mathrm{bare} \), to get the field-theoretic expression \[ E_\mathrm{field} = \frac{1}{2} \sum_i \frac{\boldsymbol{p}_i^\mathrm{bare}\boldsymbol{p}_i^\mathrm{bare}}{m_i^\mathrm{bare}} + \frac{1}{2} \int \mathrm{d}^3 \boldsymbol{x} \int \mathrm{d}^3 \boldsymbol{y} \frac{\rho(\boldsymbol{x},t)\rho(\boldsymbol{y},t)}{4\pi\epsilon_0 |\boldsymbol{x}-\boldsymbol{y}|} - \int \mathrm{d}^3 \boldsymbol{x} \; \boldsymbol{j}(\boldsymbol{x},t)\boldsymbol{A}(\boldsymbol{x},t) ,\] which is not only infinite, but uses the unphysical concept of bare mass. Comparing the expressions (4) and (10), we can easily obtain the fundamental relation between the true Coulomb energy and the field-theoretic expression —for many applications \( \rho(\boldsymbol{z};t) \) [3] can be safely replaced by \( \rho(\boldsymbol{z},t) \) because the functional difference is harmless— \[ \frac{1}{2} \sum_i\sum_{j\neq i} \frac{e_i e_j}{4\pi\epsilon_0 |\boldsymbol{r}_i -\boldsymbol{r}_j|} = \frac{1}{2} \int \mathrm{d}^3 \boldsymbol{x} \int \mathrm{d}^3 \boldsymbol{y} \frac{\rho(\boldsymbol{x},t)\rho(\boldsymbol{y},t)}{4\pi\epsilon_0 |\boldsymbol{x}-\boldsymbol{y}|} - \frac{1}{2} \sum_i \frac{e_i^2}{4\pi\epsilon_0 |\boldsymbol{r}_i -\boldsymbol{r}_i|} \] or, using a more concise notation, \[ V_\mathrm{Coulomb} = V_\mathrm{field} + V_\mathrm{countertem} .\] Of course the infinite counterterm exactly cancels the divergence within the field-theoretic expression giving a finite Coulomb energy in agreement with experiments. We have derived from the direct-particle interaction the field-theoretic interaction plus a renormalization counterterm. The existence of this counterterm is a physical consequence of the fact that the interaction between charged particles cannot be fully described in terms of interacting fields \( \rho(\boldsymbol{x},t)\phi(\boldsymbol{x},t) \). The origin of this impossibility can be traced back to the Coulomb interaction describing the physics of the correlation between particles, and this correlation is beyond a product of spacetime densities \( \rho(\boldsymbol{x},t)\rho(\boldsymbol{y},t) \). Precisely, the absence of correlations in the field-theoretic approach is the ultimate reason why a physical field is required to carry out the interaction between particles, whereas no field is needed in direct-particle-action theories.
The same line or reasoning can be applied in a quantum context, except that then one obtains also a renormalization of electron charge and a final set of quantum electrodynamic expressions in terms of bare masses and bare charges, both unphysical and different from real masses and charges. Using direct-particle gravitational interactions we can obtain general relativity expression plus renormalization corrections.
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.
[3] Sometimes the density is better written like \( \rho(\boldsymbol{z};t) \) with the semicolon indicating a time-implicit dependence through the particles positions \( \rho(\boldsymbol{z};t) = \rho(\boldsymbol{z},\{\boldsymbol{r}_i(t)\}) \).