# Just Thought You Would Like to Know

William H. Wysong iqwysong at gmail.com
Wed Nov 19 22:04:13 UTC 2014

Lagrangian formalismt
<http://en.wikipedia.org/w/index.php?title=Quantum_field_theory&action=edit&section=7>Quantum
field theory frequently makes use of the Lagrangian
<http://en.wikipedia.org/wiki/Lagrangian> formalism from classical field
theory <http://en.wikipedia.org/wiki/Classical_field_theory>. This
formalism is analogous to the Lagrangian formalism used in classical
mechanics <http://en.wikipedia.org/wiki/Classical_mechanics> to solve for
the motion of a particle under the influence of a field. In classical field
theory, one writes down a Lagrangian density
<http://en.wikipedia.org/wiki/Lagrangian_density>, [image: \mathcal{L}],
involving a field, φ(*x*,*t*), and possibly its first derivatives (∂φ/∂*t* and
∇φ), and then applies a field-theoretic form of the Euler–Lagrange equation
<http://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation>. Writing
coordinates (*t*, *x*) = (*x*0, *x*1, *x*2, *x*3) = *x*μ, this form of the
Euler–Lagrange equation is[3]
<http://en.wikipedia.org/wiki/Quantum_field_theory#cite_note-tong1-3>[image:
\frac{\partial}{\partial x^\mu}
\left[\frac{\partial\mathcal{L}}{\partial(\partial\phi/\partial
x^\mu)}\right] - \frac{\partial\mathcal{L}}{\partial\phi} = 0,]

where a sum over μ is performed according to the rules of Einstein notation
<http://en.wikipedia.org/wiki/Einstein_notation>.

By solving this equation, one arrives at the "equations of motion" of the
field.[3]
<http://en.wikipedia.org/wiki/Quantum_field_theory#cite_note-tong1-3> For
example, if one begins with the Lagrangian density
[image: \mathcal{L}(\phi,\nabla\phi) =
-\rho(t,\mathbf{x})\,\phi(t,\mathbf{x}) - \frac{1}{8\pi G}|\nabla\phi|^2,]

and then applies the Euler–Lagrange equation, one obtains the equation of
motion
[image: 4\pi G \rho(t,\mathbf{x}) = \nabla^2 \phi.]

This equation is Newton's law of universal gravitation
<http://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation>,
expressed in differential form in terms of the gravitational potential
<http://en.wikipedia.org/wiki/Gravitational_potential> φ(*t*, *x*) and the mass
density <http://en.wikipedia.org/wiki/Mass_density> ρ(*t*, *x*). Despite
the nomenclature, the "field" under study is the gravitational potential,
φ, rather than the gravitational field, *g*. Similarly, when classical
field theory is used to study electromagnetism, the "field" of interest is
the electromagnetic four-potential
<http://en.wikipedia.org/wiki/Four-potential> (*V*/*c*, *A*), rather than
the electric and magnetic fields *E* and *B*.

Quantum field theory uses this same Lagrangian procedure to determine the
equations of motion for quantum fields. These equations of motion are then
supplemented by commutation relations
<http://en.wikipedia.org/wiki/Commutation_relation> derived from the
canonical quantization procedure described below, thereby incorporating
quantum mechanical effects into the behavior of the field.

Bill Wysong,* MA, LPC, EMDR II, TEP*
*Aspen Counseling Center*