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The solutions to the equation are mathematical functions which correspond directly to the field, as functions of time and space. Since the field equation is a partial differential equation, there are families of solutions which represent a variety of physical possibilities. Usually, there is not just a single equation, but a set of coupled ...
For example, Maxwell's equations of electromagnetism are linear in the electric and magnetic fields, and charge and current distributions (i.e. the sum of two solutions is also a solution); another example is Schrödinger's equation of quantum mechanics, which is linear in the wavefunction.
A field theory tends to be expressed mathematically by using Lagrangians. This is a function that, when subjected to an action principle, gives rise to the field equations and a conservation law for the theory. The action is a Lorentz scalar, from which the field equations and symmetries can be readily derived.
Defining equation SI unit Dimension Wavefunction: ψ, Ψ To solve from the Schrödinger equation: varies with situation and number of particles Wavefunction probability density: ρ = | | = m −3 [L] −3: Wavefunction probability current: j: Non-relativistic, no external field:
Examples include the gravitational field in Newtonian gravity g(x, t) and the electric field E(x, t) and magnetic field B(x, t) in classical electromagnetism. A classical field can be thought of as a numerical quantity assigned to every point in space that changes in time. Hence, it has infinitely many degrees of freedom. [29] [30]
These tensor fields should obey any relevant physical laws (for example, any electromagnetic field must satisfy Maxwell's equations).Following a standard recipe which is widely used in mathematical physics, these tensor fields should also give rise to specific contributions to the stress–energy tensor. [1]
In field theory, the independent variable is replaced by an event in spacetime (x, y, z, t), or more generally still by a point s on a Riemannian manifold.The dependent variables are replaced by the value of a field at that point in spacetime (,,,) so that the equations of motion are obtained by means of an action principle, written as: =, where the action, , is a functional of the dependent ...
This is clearly not enough, as there are only 14 equations (10 from the field equations and 4 from the continuity equation) for 20 unknowns (10 metric components and 10 stress–energy tensor components). Equations of state are missing. In the most general case, it's easy to see that at least 6 more equations are required, possibly more if ...