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Field equations can be classified in many ways: classical or quantum, nonrelativistic or relativistic, according to the spin or mass of the field, and the number of components the field has and how they change under coordinate transformations (e.g. scalar fields, vector fields, tensor fields, spinor fields, twistor fields etc.).
As the term suggests, an EM field consists of two vector fields, an electric field (,) and a magnetic field (,).Both are time-dependent vector fields that in vacuum depend on a third vector field (,) (the vector potential), as well as a scalar field (,)
A field can be classified as a scalar field, a vector field, a spinor field or a tensor field according to whether the represented physical quantity is a scalar, a vector, a spinor, or a tensor, respectively. A field has a consistent tensorial character wherever it is defined: i.e. a field cannot be a scalar field somewhere and a vector field ...
One can express the complex scalar field theory in terms of two real fields, φ 1 = Re φ and φ 2 = Im φ, which transform in the vector representation of the U(1) = O(2) internal symmetry. Although such fields transform as a vector under the internal symmetry , they are still Lorentz scalars.
Divergence is a vector operator that produces a signed scalar field giving the quantity of a vector field's source at each point. Note that in this metric signature [+,−,−,−] the 4-Gradient has a negative spatial component. It gets canceled when taking the 4D dot product since the Minkowski Metric is Diagonal[+1,−1,−1,−1].
Consider a generic (possibly non-Abelian) gauge transformation acting on a component field = =.The main examples in field theory have a compact gauge group and we write the symmetry operator as () = where () is an element of the Lie algebra associated with the Lie group of symmetry transformations, and can be expressed in terms of the hermitian generators of the Lie algebra (i.e. up to a ...
Nordström's theories, on the other hand, are scalar theories because the gravitational field is a scalar. Other proposed alternatives include scalar–tensor theories that contain a scalar field in addition to the tensors of general relativity, and other variants containing vector fields as well have been developed recently.
The force on a test particle subject only to gravity and electromagnetism is = +, where p α is the linear 4-momentum of the particle, t is any time coordinate parameterizing the world line of the particle, Γ β αγ is the Christoffel symbol (gravitational force field), and q is the electric charge of the particle.