Parenthesis on the Minimal Interaction

The study of the movement of charged particles in an electromagnetic field exemplifies the treatment of a non-conservative system as the potential depends on the velocity. Nevertheless, for this system the equations of Euler-Lagrange can be written

$$\frac{d}{dt} (\frac{\partial L}{\partial \dot{q}_{j}}) - \frac{\partial L}{\partial q_{j}} = 0$$ (127)

with

$$L = T - U$$ (128)

and $U$, the generalized potential which gives place to the generalized forces

$$Q_{j} = - \frac{\partial U}{\partial q_{j}} + \frac{d}{dt} (\frac{\partial U}{\partial \dot{q}_{j}}) = 0$$ (129)

as

$$\vec{F} = q [\vec{E} + \frac{1}{c} (\vec{v}\wedge \vec{B})]$$ (130)

it can be written

$$F_{i} = \frac{\partial U}{\partial x_{i}} + \frac{d}{dt} (\frac{\partial U}{\partial v_{i}})$$ (131)

with

$$U = q \Phi - \frac{q}{c} \vec{A} \cdot \vec{v}$$ (132)

in such a way that

$$L = T - q \Phi + \frac{q}{c} \vec{A} \cdot \vec{v}$$ (133)

>From the knowledge of $L$, the Hamiltonian formalism can be followed but with no possible way of insuring a priori that $H$ coincides with the total energy. Nevertheless, that is the case for the electromagnetism (why?).

From $L$, the canonical momentum results

$$p_{i} = m v_{i} + \frac{q}{c} A_{i}$$ (134)

and then

$$H = m v^{2} + \frac{q}{c} \vec{A} \cdot \vec{v} - L = T + q \Phi = E$$ (135)

The Hamiltonian can be written then in terms of the canonical momentum as

$$H = \frac{1}{2 m} \left[\vec{p} - \frac{q}{c} \vec{A}\right]^{2} + q \Phi$$ (136)

that is also a valid form for the relativistic case.

A recipe can then be formalized: The Hamiltonian of a charged particle in an electromagnetic field is obtained from the case for a generic $V$ potential, through the replacement

$$V \rightarrow q \Phi$$ (137)

$$canonical\; momentum:\; \vec{p} \rightarrow \vec{p} - \frac{q}{c} \vec{A}$$ (138)

which is called minimal interaction.

The quantum mechanics of a charged particle in an electromagnetic field is obtained, $\grave{a}\; la$ Schrödinger, translating operationally the Hamiltonian written in terms of the coordinate and the canonical momentum. In that way the Schrödinger equation arises

$$i \hbar \frac{\partial \psi}{\partial t}(\vec{r},t) = \lef... ...ac{q}{c} \vec{A}\right)^{2} + q \Phi\right] \psi(\vec{r},t)$$ (139)

For QED, theory of electrons and photons, the Lagrangian is

$${\cal L} = {\cal L}_{em} + {\cal L}_{e} + {\cal L}_{I}$$ (140)

with

$${\cal L}_{0} = {\cal L}_{em} + {\cal L}_{e}$$ (141)

$$= - \frac{1}{4} F_{\mu\nu} F^{\mu\nu} + \bar{\psi} (i \gamma^{\mu}\partial_{\mu} - m) \psi$$ (142)

and

$${\cal L}_{I} = - e \bar{\psi} \gamma_{\mu} A^{\mu} \psi \;\;\; [minimal]$$ (143)

Again, the minimal electromagnetic interaction comes from replacing in the equation of motion of the electron

$$\partial_{\mu} \rightarrow \partial_{\mu} - i e A_{\mu}$$ (144)

This replacement is entirely analogous to the one of the quantum mechanics and gives place to a renormalizable theory. We could certainly think of a more general interaction (not minimal). For example of the type

$$e' \bar{\psi} \sigma_{\mu\nu} \psi F^{\mu\nu} \nonumber$$

which has the required properties but gives rise to a non-renormalizable theory.

It will be then a challenge to provide a theoretical background for the presence of the minimal interaction in the electromagnetic case and try to generalize the idea to the other interactions. The recent developments are based on symmetry requierements to be fullfilled by the theoretical structure. This is the origin of Gauge Theories.