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Electromagnetic tensor

The electromagnetic tensor, also known as the tensor, is a second-rank in the theory of that unifies the \mathbf{E} and \mathbf{B} into a single mathematical object, enabling a covariant formulation of electrodynamics under Lorentz transformations. It is defined as F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu, where A^\mu = (\phi/c, \mathbf{A}) is the four-potential consisting of the \phi and \mathbf{A}. This tensor has six independent components, corresponding to the three components each of \mathbf{E} and \mathbf{B}, and its antisymmetry ensures F^{\mu\nu} = -F^{\nu\mu} with zero diagonal elements. In the standard matrix representation using the Minkowski metric with signature (+,-,-,-), the contravariant tensor F^{\mu\nu} takes the form: F^{\mu\nu} = \begin{pmatrix} 0 & -E_x/c & -E_y/c & -E_z/c \\ E_x/c & 0 & -B_z & B_y \\ E_y/c & B_z & 0 & -B_x \\ E_z/c & -B_y & B_x & 0 \end{pmatrix}, where the upper indices run over \mu,\nu = 0,1,2,3 corresponding to time and spatial coordinates. The electric field components appear in the time-space elements (F^{0i} = -E_i/c), while the magnetic field components fill the spatial-spatial block (F^{ij} = -\epsilon^{ijk} B_k), with \epsilon^{ijk} the Levi-Civita symbol. The lowered-index version F_{\mu\nu} follows from raising and lowering with the metric tensor, adjusting signs accordingly. This tensor plays a central role in relativistic electrodynamics by transforming covariantly under Lorentz boosts, mixing \mathbf{E} and \mathbf{B} fields—for instance, a pure in one frame may appear as a combination of electric and magnetic fields in a boosted frame, as in \mathbf{E}'_\parallel = \mathbf{E}_\parallel and \mathbf{E}'_\perp = \gamma (\mathbf{E}_\perp + \mathbf{v} \times \mathbf{B}_\perp). in tensor form are compactly expressed as \partial_\mu F^{\mu\nu} = \mu_0 J^\nu for the inhomogeneous equations and \partial_\lambda F_{\mu\nu} + \partial_\mu F_{\nu\lambda} + \partial_\nu F_{\lambda\mu} = 0 for the homogeneous ones, where J^\nu is the four-current. Additionally, the on a is given by m \frac{du^\mu}{d\tau} = q F^\mu{}_\nu u^\nu, highlighting its utility in describing dynamics in electromagnetic fields. The tensor also extends naturally to and , where it underlies the gauge-invariant description of .

Definition and Components

Relation to Classical Fields

In , the \mathbf{E} and \mathbf{B} are treated as separate three-dimensional fields, with describing their dynamics in a non-relativistic framework. However, reveals that \mathbf{E} and \mathbf{B} are interdependent components of a single entity, as Lorentz transformations mix them: an in one inertial frame appears as a combination of electric and s in another frame moving relative to the first. This mixing necessitates a unified description to ensure the laws of remain invariant under Lorentz transformations, contrasting the frame-dependent formulations of classical theory. The electromagnetic field tensor F^{\mu\nu} provides this unification by encoding both \mathbf{E} and \mathbf{B} within a single antisymmetric 4×4 tensor in four-dimensional Minkowski . Introduced by in as part of his formalism for in electrodynamics, the tensor F^{\mu\nu} captures the six independent components of the —three from \mathbf{E} and three from \mathbf{B}—in a Lorentz-covariant manner. This structure arose from Minkowski's efforts to reformulate in the context of his newly developed geometry, ensuring consistency with the principle of relativity. The tensor's antisymmetry (F^{\mu\nu} = -F^{\nu\mu}) naturally accommodates the field's properties, while its transformation under Lorentz boosts and rotations preserves the physical content across frames. The spatial components of the tensor relate to the through the \epsilon^{ijk}, with F^{ij} = -\sum_k \epsilon^{ijk} B^k (in units where c=1), where i,j,k = 1,2,3. The time-space components link to the as F^{0i} = E^i. These relations demonstrate how F^{\mu\nu} organizes the classical fields into a relativistic object, where the off-diagonal blocks of the 4×4 matrix distinguish electric and magnetic contributions while maintaining overall covariance. This encoding facilitates the derivation of Lorentz-invariant quantities, such as the scalar F_{\mu\nu} F^{\mu\nu}, which motivates the tensor's use by combining \mathbf{E}^2 - \mathbf{B}^2 into a frame-independent expression.

Explicit Components and Conventions

The electromagnetic tensor F^{\mu\nu} is a rank-2 in four-dimensional , with components expressed in Cartesian coordinates using the mostly plus \eta^{\mu\nu} = \operatorname{diag}(1, -1, -1, -1). Its diagonal elements are zero due to antisymmetry, F^{\mu\nu} = -F^{\nu\mu}, leaving six independent components that correspond to the three components E^i and three components B_i. In , the explicit form is given by F^{0i} = E^i / c and F^{i0} = -E^i / c for the time-space components, while the space-space components are F^{ij} = -\epsilon^{ijk} B_k, where \epsilon^{ijk} is the and summation over k is implied. The full 4×4 matrix representation of F^{\mu\nu} is: \begin{pmatrix} 0 & E_x/c & E_y/c & E_z/c \\ -E_x/c & 0 & -B_z & B_y \\ -E_y/c & B_z & 0 & -B_x \\ -E_z/c & -B_y & B_x & 0 \end{pmatrix} This structure ensures the tensor's antisymmetry and encodes the classical electric and magnetic fields as its off-diagonal elements. The covariant tensor F_{\mu\nu} is obtained by lowering the indices with the , F_{\mu\nu} = \eta_{\mu\alpha} \eta_{\nu\beta} F^{\alpha\beta}, which introduces sign flips for spatial components due to the negative spatial eigenvalues. Explicitly, F_{0i} = -E_i / c and F_{i0} = E_i / c, while F_{ij} = -\epsilon_{ijk} B^k. The resulting matrix for F_{\mu\nu} is: \begin{pmatrix} 0 & -E_x/c & -E_y/c & -E_z/c \\ E_x/c & 0 & -B_z & B_y \\ E_y/c & B_z & 0 & -B_x \\ E_z/c & -B_y & B_x & 0 \end{pmatrix} Conventions for the metric signature vary across treatments; while the above uses (+,-,-,-), the opposite (-,+,+,+) signature reverses the signs of the time-space components, yielding F^{0i} = -E^i / c and F^{i0} = E^i / c in the matrix, with spatial components unchanged. In natural units common to high-energy physics, c = 1 and often \hbar = 1, simplifying the expressions to F^{0i} = E^i without the /c factors. These variations maintain the physical equivalence of the fields but require careful adjustment in calculations.

Mathematical Properties

Antisymmetry and Structure

The electromagnetic field tensor F^{\mu\nu} exhibits antisymmetry, satisfying F^{\mu\nu} = -F^{\nu\mu}, which arises directly from its definition as the curl of the four-potential: F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu. This property ensures that swapping the indices inverts the sign, reflecting the inherent orientation reversal in the underlying . As a contravariant tensor of rank (2,0), F^{\mu\nu} possesses 16 components in four-dimensional , but antisymmetry reduces the number of independent components to six, since the diagonal elements vanish and off-diagonal pairs are related by negation. Similarly, its covariant counterpart F_{\mu\nu} is a rank (0,2) tensor with the same constraints. A key implication of this antisymmetry is the vanishing trace, F^\mu_\mu = 0, as the summation over indices pairs each component with its negative counterpart. Furthermore, in different Lorentz frames, the tensor decomposes into distinct parts that correspond to electric and magnetic contributions, highlighting its frame-dependent structure while preserving the total independent degrees of freedom. Geometrically, the electromagnetic tensor represents an antisymmetric on , equivalent to a differential 2-form in the framework of . This interpretation underscores its role as a skew-symmetric object that encodes the field's orientable properties, facilitating over surfaces and with the in form calculus.

Invariants and Dual Tensor

The electromagnetic tensor F^{\mu\nu} gives rise to two primary Lorentz-invariant scalars, which are frame-independent quantities that characterize the field's properties across all inertial observers. These invariants are the F_{\mu\nu} F^{\mu\nu} and the F_{\mu\nu} {}^*F^{\mu\nu}, where {}^*F^{\mu\nu} is the tensor defined below. In where c = 1, the first invariant evaluates to F_{\mu\nu} F^{\mu\nu} = 2(B^2 - E^2), distinguishing electric-dominated fields (negative value) from magnetic-dominated ones (positive value). The second invariant is F_{\mu\nu} {}^*F^{\mu\nu} = -4 \mathbf{E} \cdot \mathbf{B}, which vanishes for perpendicular electric and magnetic fields and serves as a measure of their . These scalars are invariant under Lorentz transformations because F^{\mu\nu} transforms as a rank-2 tensor, F'^{\mu\nu} = \Lambda^\mu{}_\rho \Lambda^\nu{}_\sigma F^{\rho\sigma}, where \Lambda is the matrix with \det \Lambda = 1. Contracting indices with the g^{\mu\nu} (raised with the Minkowski metric) yields quantities with no free indices, which remain unchanged regardless of the frame. Similarly, the preserves its value due to the proper orthogonal nature of Lorentz transformations, ensuring the overall expression is scalar. The dual tensor {}^*F^{\mu\nu} is defined as {}^*F^{\mu\nu} = \frac{1}{2} \epsilon^{\mu\nu\rho\sigma} F_{\rho\sigma}, where \epsilon^{\mu\nu\rho\sigma} is the Levi-Civita symbol with \epsilon^{0123} = +1. This antisymmetric pseudo-tensor interchanges the roles of electric and magnetic components: its spatial parts correspond to the electric field, while the time-space parts relate to the magnetic field. A key property in four-dimensional Minkowski spacetime is that applying the dual operation twice yields {}^{**}F^{\mu\nu} = -F^{\mu\nu}, arising from the contraction of two Levi-Civita symbols and the antisymmetry of F^{\mu\nu}. Physically, these invariants underpin covariant descriptions of electromagnetic phenomena, such as the radiation reaction force on accelerating charges in the Abraham-Lorentz-Dirac equation, where the field's type (determined by the signs and values of the invariants) influences the self-force magnitude. They also appear in the covariant form of , which expresses through the symmetric energy-momentum tensor T^{\mu\nu} = F^{\mu\lambda} F^\nu{}_\lambda - \frac{1}{4} g^{\mu\nu} F_{\rho\sigma} F^{\rho\sigma}, whose divergence vanishes identically from . The dual tensor relates hypothetically to magnetic monopoles, as introducing a magnetic charge current J^\mu_m modifies to \partial_\mu {}^*F^{\mu\nu} = J^\nu_m, symmetrizing the theory and restoring electric-magnetic duality.

Relativistic Formulation

Covariant Description

In the framework of , the is described by the rank-2 F^{\mu\nu}, which unifies the \mathbf{E} and \mathbf{B} into a single entity that transforms covariantly under Lorentz transformations. This tensorial formulation ensures that the laws of are invariant across inertial frames, capturing the interdependence of \mathbf{E} and \mathbf{B}. The transformation law for the electromagnetic tensor under a \Lambda^\mu{}_\rho is given by F'^{\mu\nu} = \Lambda^\mu{}_\alpha \Lambda^\nu{}_\beta F^{\alpha\beta}, where the primed tensor F'^{\mu\nu} denotes the components in the transformed frame. This tensor transformation preserves the antisymmetry of F^{\mu\nu} and the Lorentz invariants, such as F_{\mu\nu} F^{\mu\nu} and the F_{\mu\nu} {}^*F^{\mu\nu}. Under a Lorentz boost along the x-direction with velocity v, the parallel components remain unchanged: E'_\parallel = E_\parallel and B'_\parallel = B_\parallel, while the perpendicular components mix as E'_\perp = \gamma (E_\perp + v \times B)_\perp and B'_\perp = \gamma (B_\perp - (v/c^2) \times E)_\perp, with \gamma = 1/\sqrt{1 - v^2/c^2}. For instance, a pure in one frame may appear as a combination of electric and s in the boosted frame, illustrating the relativistic unification of electricity and . The Maxwell equations take a compact covariant form using F^{\mu\nu}: the inhomogeneous equation \partial_\mu F^{\mu\nu} = \mu_0 J^\nu relates the field to the four-current J^\nu, while the homogeneous equation \partial_\mu {}^*F^{\mu\nu} = 0 (with {}^*F^{\mu\nu} = \frac{1}{2} \epsilon^{\mu\nu\rho\sigma} F_{\rho\sigma} the tensor) encodes the field . These expressions are manifestly Lorentz invariant, eliminating the need for separate vector equations for \mathbf{E} and \mathbf{B} and simplifying the treatment of relativistic effects.

Connection to Four-Potential

The electromagnetic field tensor F^{\mu\nu} is derived from the four-potential A^\mu, which combines the \phi and the \mathbf{A} into a single A^\mu = (\phi/c, \mathbf{A}). The tensor is defined as the antisymmetric difference of partial derivatives: F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu. This construction ensures that F^{\mu\nu} transforms as a rank-2 tensor under Lorentz transformations, capturing both electric and magnetic fields in a covariant manner. A key feature of this definition is its invariance: under the A^\mu \to A^\mu + \partial^\mu \chi, where \chi is an arbitrary scalar function, the field tensor remains unchanged because the added terms cancel due to the antisymmetry: F^{\mu\nu} \to (\partial^\mu (A^\nu + \partial^\nu \chi) - \partial^\nu (A^\mu + \partial^\mu \chi)) = F^{\mu\nu} + (\partial^\mu \partial^\nu - \partial^\nu \partial^\mu) \chi = F^{\mu\nu}. This property allows physical observables, encoded in F^{\mu\nu}, to be independent of the choice of gauge for the potentials. In the classical limit, the components of F^{\mu\nu} recover the familiar expressions for the electric and magnetic fields from the potentials. Specifically, the electric field arises as \mathbf{E} = -\nabla \phi - \partial \mathbf{A}/\partial t, while the magnetic field is given by \mathbf{B} = \nabla \times \mathbf{A}; these are obtained by extracting the appropriate tensor components in the (+,-,-,-) metric convention, such as F^{0i} = E^i/c and F^{ij} = -\epsilon^{ijk} B^k. The definition of F^{\mu\nu} from the four-potential also implies a identity known as the Bianchi identity: \partial_\lambda F_{\mu\nu} + \partial_\mu F_{\nu\lambda} + \partial_\nu F_{\lambda\mu} = 0. This identity holds automatically due to the equality of mixed partial derivatives and the antisymmetry of F_{\mu\nu}, and in three-vector form, it corresponds to the classical relations \nabla \cdot \mathbf{B} = 0 and \partial \mathbf{B}/\partial t = -\nabla \times \mathbf{E}. The Bianchi identity represents the homogeneous set of , independent of sources.

Applications in Field Theories

Lagrangian and Hamiltonian Approaches

In the Lagrangian formulation of classical electromagnetism, the electromagnetic tensor F_{\mu\nu} serves as the fundamental dynamical object through the four-potential A_\mu, from which F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu. The Lagrangian density is given by \mathcal{L} = -\frac{1}{4} F_{\mu\nu} F^{\mu\nu} - A_\mu J^\mu, in units where c = 1 and \epsilon_0 = 1, where J^\mu is the four-current density. This form is gauge-invariant under A_\mu \to A_\mu + \partial_\mu \Lambda and leads to the equations of motion via the Euler-Lagrange equations for A_\nu: \partial_\mu \left( \frac{\partial \mathcal{L}}{\partial (\partial_\mu A_\nu)} \right) - \frac{\partial \mathcal{L}}{\partial A_\nu} = 0, which simplify to \partial_\mu F^{\mu\nu} = J^\nu, recovering the inhomogeneous Maxwell equations; the homogeneous equations \partial_{[\lambda} F_{\mu\nu]} = 0 follow from the definition of F_{\mu\nu}. The conservation of energy and momentum in the electromagnetic field arises from the translational invariance of the Lagrangian via Noether's theorem, yielding the symmetric stress-energy-momentum tensor T^{\mu\nu} = F^{\mu\lambda} F^\nu{}_\lambda - \frac{1}{4} \eta^{\mu\nu} F_{\rho\sigma} F^{\rho\sigma}, where \eta^{\mu\nu} is the Minkowski metric (signature (+,-,-,-)). In the presence of sources, the divergence satisfies \partial_\mu T^{\mu\nu} = -F^{\nu\lambda} J_\lambda, expressing the transfer of energy-momentum from the field to matter; for free fields, \partial_\mu T^{\mu\nu} = 0 implies local conservation. The time-time component T^{00} corresponds to the field energy density \frac{1}{2}(E^2 + B^2), while the spatial components encode momentum density and stresses. The Hamiltonian formulation proceeds by Legendre transformation from the Lagrangian, treating the transverse components of A_i (in Coulomb gauge) as coordinates. The canonical momenta are \pi^i = \frac{\partial \mathcal{L}}{\partial (\partial_0 A_i)} = -E^i, with the scalar potential \phi as a Lagrange multiplier enforcing the constraint \nabla \cdot \mathbf{E} = \rho. The Hamiltonian is then H = \int \frac{1}{2} (E^2 + B^2) \, d^3x, which generates the field dynamics via Hamilton's equations, \partial_0 A_i = \frac{\delta H}{\delta \pi^i} and \partial_0 \pi^i = -\frac{\delta H}{\delta A_i}, reproducing Faraday's and Ampère's laws. In a covariant phase space approach, the formulation extends to relativistic constraints, incorporating the full tensor structure while preserving gauge invariance. The tensor form of follows directly from the divergence of T^{\mu\nu}, providing a covariant expression for energy-momentum conservation: the term -F^{\nu\lambda} J_\lambda represents the density on charges, linking field transport to mechanical work. This unifies the classical \mathbf{S} = \mathbf{E} \times \mathbf{B} (as T^{0i}) with relativistic extensions.

Role in

In (), the electromagnetic tensor F^{\mu\nu} serves as the fundamental operator describing the quantized field, bridging to the quantum regime of interacting fermions and gauge bosons. The quantization procedure elevates the classical tensor to an operator by first quantizing the four-potential A^\mu in the Lorentz gauge, \partial_\mu A^\mu = 0, where the field strength follows as F^{\mu\nu}(x) = \partial^\mu A^\nu(x) - \partial^\nu A^\mu(x). This yields canonical commutation relations for the fields, such as [A^\mu(x), \dot{A}^\nu(y)] = -i g^{\mu\nu} \delta^{(3)}(\mathbf{x} - \mathbf{y}) at equal times, ensuring the tensor satisfies the operator algebra consistent with the Maxwell equations in vacuum. Specifically, commutation relations involving the tensor take the form [A^\mu(x), \partial_\nu F^{\nu\lambda}(y)] \sim \delta^\mu_\lambda \delta^{(4)}(x - y), reflecting the field's role in generating currents and enforcing invariance in the quantum theory. The QED Lagrangian incorporates the electromagnetic tensor through its classical kinetic term, extended to include fermionic interactions: \mathcal{L}_\text{QED} = \bar{\psi} (i \gamma^\mu D_\mu - m) \psi - \frac{1}{4} F_{\mu\nu} F^{\mu\nu}, where the D_\mu = \partial_\mu - i e A_\mu couples electrons to the photon field, and e is the . This structure highlights F^{\mu\nu}'s origin in the free-field limit of classical electrodynamics, now quantized, with the tensor's quadratic term deriving from the gauge-invariant for the . Upon quantization, the interaction term generates perturbative expansions via Feynman diagrams, where the rules specify the vertex factor for the fermion-photon coupling as -i e \gamma^\mu. The propagator, arising as the inverse of the kinetic operator encoded in the F_{\mu\nu} F^{\mu\nu} term (in Feynman gauge, i D^{\mu\nu}(k) = -i g^{\mu\nu}/k^2), facilitates the summation of virtual photon exchanges in scattering amplitudes. Applications of the electromagnetic tensor in prominently feature in higher-order corrections, such as , where loops modify the photon self-energy \Pi^{\mu\nu}(q), effectively renormalizing the tensor's . In photon self-energy diagrams, the loop involves two F^{\mu\nu} vertices connected by propagators, leading to a transverse structure \Pi^{\mu\nu}(q) = (q^2 g^{\mu\nu} - q^\mu q^\nu) \Pi(q^2) that preserves gauge invariance and alters the effective charge at short distances. These effects necessitate renormalization of the QED invariants, particularly F_{\mu\nu} F^{\mu\nu} and its dual, through counterterms that absorb divergences in the photon wave function renormalization constant Z_3, ensuring finite predictions for observables like the anomalous . Such processes underscore F^{\mu\nu}'s pivotal role in maintaining the theory's renormalizability and .

References

  1. [1]
    The Electromagnetic Field Tensor
    The Electromagnetic Field Tensor. The transformation of electric and magnetic fields under a Lorentz boost we established even before Einstein developed the ...
  2. [2]
    Rel Electrodynamics - Galileo and Einstein
    How does this work in four dimensions? To find out, we define the electromagnetic field tensor. Fμν=∂μAν−∂νAμ. (This is the standard notation, with both ...
  3. [3]
    [PDF] 5. THE ELECTROMAGNETIC FIELD TENSOR
    The anti-symmetric tensor Fµν is called the electromagnetic field tensor; its components will be detailed shortly. • Eq. (7) is the covariant form of the ...
  4. [4]
    [PDF] 5. Electromagnetism and Relativity - DAMTP
    Under Lorentz transformations, electric and magnetic fields will transform into each other. In this section, we want to understand more precisely how this ...
  5. [5]
    [PDF] The Classical Theory of Fields
    ... Electromagnetic ... for their assistance in checking formulas. Moscow, September 1961. L. D. Landau, E. M. Lifshitz. Page 11. PREFACE. TO THE THIRD ENGLISH ...
  6. [6]
    [PDF] 12.1 The field equations - MIT
    More specifically, we found that the spatial components of dpα/dτ correspond exactly to the 3-force F = q(E+v×B), and that the 0 or timelike component tells us ...
  7. [7]
    The Electromagnetic Field Tensor
    The electromagnetic field tensor combines electric and magnetic fields into a rank 2 antisymmetric tensor, needed for Lorentz transformations.
  8. [8]
    [PDF] Reformulation of Electromagnetism with Differential Forms
    It has been shown that the four-dimensional differential forms (anti-symmetric covariant tensors) are the most suitable tools for expressing the structure of ...
  9. [9]
    The dual electromagnetic field tensor - Richard Fitzpatrick
    The dual electromagnetic field tensor. We have seen that it is possible to write the components of the electric and magnetic fields as the components of a ...<|control11|><|separator|>
  10. [10]
    [PDF] On Electromagnetic Duality
    Nov 14, 2018 · These notes are intended to clarify what is meant by dual tensors and to shed some light upon the group of such duality transformations. II.
  11. [11]
    Charged particle motion and radiation in strong electromagnetic fields
    Oct 7, 2022 · This review explores the basic physical processes of radiation reaction and QED in strong fields, how they are treated theoretically and in simulation.<|separator|>
  12. [12]
    [PDF] A derivation of Poynting's Theorem in relativistic electromagnetism
    It gives the isomorphism between contravariant vectors and covariant dual-vectors. ... If this equation holds in one frame of reference, then by the invariance of ...
  13. [13]
    [PDF] Abelian Gauge Fields
    The dual equation is an identity (Bianchi identity) if F is constructed from a vector potential a: Fµν = ∂µAν. − ∂. νAµ, Aµ = (φ, A). (2.12). The Lorentz ...
  14. [14]
    The Lagrangian for Electromagnetic Fields
    The Lagrangian for Electromagnetic Fields. There are not many ways to make a scalar Lagrangian from the field tensor. We already know that. \begin{ ...Missing: F^ | Show results with:F^
  15. [15]
    [PDF] Stress-Energy tensor for Maxwell Theory - Rutgers Physics
    While this Tνµ is a correct conserved current even if we drop the Λ term, there are two unpleasant features. First, it is not symmetric under µ ↔ ν, which we ...
  16. [16]
    [PDF] The Classical Electromagnetic Field Hamiltonian
    In these notes we motivate the necessity for quantizing the electromagnetic field, we outline a procedure for doing that, and then we carry out the ...
  17. [17]
    None
    ### Hamiltonian Formulation for the Electromagnetic Field
  18. [18]
    [PDF] 6. Quantum Electrodynamics - DAMTP
    6.2 The Quantization of the Electromagnetic Field. In the following we shall quantize free Maxwell theory twice: once in Coulomb gauge, and again in Lorentz ...Missing: tensor | Show results with:tensor
  19. [19]
    [PDF] The Quantized Electromagnetic Field
    As we say, we have found a representation of the commutation relations (1) by means of specific operators acting on a specific Hilbert space. This procedure is ...
  20. [20]
    [PDF] Nonlinear Electrodynamics and QED - arXiv
    This paper reviews the limits of linear electrodynamics, explores nonlinear extensions, and uses QED's successes to guide understanding of nonlinear ...
  21. [21]
    [PDF] QED Feynman rules - UT Physics
    Here we have used the Feynman gauge for the photon propagator, but any other gauge would produce exactly the same amplitude. iM = ¯u′. 1(ieγµ)u1 × ¯u′. 2(ieγν) ...
  22. [22]
    Quantum Electrodynamics. II. Vacuum Polarization and Self-Energy
    It is inferred that the electromagnetic energy-momentum tensor, and the current vector must vanish in the vacuum, while the matter field energy-momentum tensor ...
  23. [23]
    [PDF] 3 One-Loop Counterterms in QED - 3.1 Fermion Self-energy
    2. Using the Feynman rules the vertex correction factor, Γµ(p, p∨) is given (in d-dimensions and in. Feynman gauge) by. Γµ(p, p∨) = µ4−d / ddk. (2π)d ...<|separator|>
  24. [24]
    [PDF] 11 Renormalization of QED - CFTP
    11 Renormalization of QED. 105. Likewise, the only two possible tensors for the photon vacuum polarization are gµν and qµqν, so we can write. Πµν(q) = a(q2)gµν ...