Field theory coupling of charge but not higher moments
In
Electrodynamics
In
electrodynamics, minimal coupling is adequate to account for all electromagnetic interactions. Higher moments of particles are consequences of minimal coupling and non-zero
spin.
Non-relativistic charged particle in an electromagnetic field
In
SI Units
):
where q is the electric charge of the particle, φ is the electric scalar potential, and the Ai, i = 1, 2, 3, are the components of the magnetic vector potential that may all explicitly depend on and .
This Lagrangian, combined with Euler–Lagrange equation, produces the Lorentz force law
and is called minimal coupling.
Note that the values of scalar potential and vector potential would change during a gauge transformation,[2] and the Lagrangian itself will pick up extra terms as well, but the extra terms in the Lagrangian add up to a total time derivative of a scalar function, and therefore still produce the same Euler–Lagrange equation.
The
canonical momenta
are given by
Note that canonical momenta are not
kinetic momenta
are gauge invariant and physically measurable.
The Hamiltonian, as the Legendre transformation of the Lagrangian, is therefore
This equation is used frequently in quantum mechanics.
Under a gauge transformation,
where f(r,t) is any scalar function of space and time, the aforementioned Lagrangian, canonical momenta and Hamiltonian transform like
which still produces the same Hamilton's equation:
In quantum mechanics, the
U(1) group transformation
[3] during the gauge transformation, which implies that all physical results must be invariant under local U(1) transformations.
Relativistic charged particle in an electromagnetic field
The relativistic Lagrangian for a particle (rest mass m and charge q) is given by:
Thus the particle's canonical momentum is
that is, the sum of the kinetic momentum and the potential momentum.
Solving for the velocity, we get
So the Hamiltonian is
This results in the force equation (equivalent to the Euler–Lagrange equation)
from which one can derive
The above derivation makes use of the vector calculus identity:
An equivalent expression for the Hamiltonian as function of the relativistic (kinetic) momentum, P = γmẋ(t) = p - qA, is
This has the advantage that kinetic momentum P can be measured experimentally whereas canonical momentum p cannot. Notice that the Hamiltonian (
,
V = eφ.
Inflation
In studies of
inflaton field
is not coupled to the
scalar curvature. Its only coupling to gravity is the coupling to the
Lorentz invariant measure constructed from the
metric (in
Planck units):
where , and utilizing the gauge covariant derivative.
References