Hamilton's principle/Applications/Classical field theory

Plain-language version: Fields

Classical field theory is the variational method applied to systems whose degrees of freedom are labelled by position in space rather than by an index. The vibrating string of Continuum systems: the vibrating string is the one-dimensional case; everything structural was already visible there. This page states the general construction and the two features that only appear once there is more than one space dimension: relativistic invariance, and the stress-energy tensor.

The construction

Local conservation as a flow across a boundary.

A field is a function ${\varphi }_i(x)$ of the spacetime coordinates $x=(x^0,x^1,x^2,x^3)$. The action is the integral of a Lagrangian density over a spacetime region:

$S[\varphi ]={\int }_{\Omega }ℒ({\varphi }_i,{\partial }_{\alpha }{\varphi }_i,x)d^{4}x\mathrm{(}\mathrm{1}\mathrm{)}$

By Several independent variables, the stationarity condition with variations vanishing on $\partial \Omega$ is

$\frac{\partial ℒ}{\partial {\varphi }_i}-{\partial }_{\alpha }\left(\frac{\partial ℒ}{\partial ({\partial }_{\alpha }{\varphi }_i)}\right)=0\mathrm{(}\mathrm{2}\mathrm{)}$

one equation per field. That is the whole formalism. What makes it powerful is not (2) but the constraints one can put on $ℒ$ before writing it down.

Locality and its consequences

Locality: influence passing from neighbour to neighbour.

Requiring $ℒ$ to depend on the fields at a single point and on finitely many derivatives there is the assumption of locality. Two consequences follow immediately.

Field equations are partial differential equations, not integral equations. Influence propagates through neighbouring points rather than acting at a distance.

Conservation laws are local. Noether's theorem on Noether's first theorem gives ${\partial }_{\alpha }{j}^{\alpha }=0$ — a density and a current obeying a continuity equation — rather than a global constant. A quantity does not simply stay constant; it flows. In one space dimension this was the energy density and flux (8) of the string page.

Restricting to first derivatives is a further choice, and it is not innocent: it makes (2) second order, which is what keeps the initial-value problem well posed. Higher derivatives generally produce instabilities, which is why essentially every field theory in use is first-order in $ℒ$. The Einstein-Hilbert action is the significant exception, and it escapes the problem for a reason specific to it.

Building a Lagrangian from symmetry

The practical method is to write the most general local density consistent with the symmetries required, and let (2) produce the dynamics.

For a real scalar field with Lorentz invariance, the available scalars from $\varphi$ and one derivative are $\varphi$ itself and ${\partial }_{\alpha }\varphi {\partial }^{\alpha }\varphi$. So

$ℒ=\frac{1}{2}{\partial }_{\alpha }\varphi {\partial }^{\alpha }\varphi -V(\varphi )\mathrm{(}\mathrm{3}\mathrm{)}$

is the general form. Applying (2),

${\partial }_{\alpha }{\partial }^{\alpha }\varphi +V^{\prime }(\varphi )=0\mathrm{(}\mathrm{4}\mathrm{)}$

With $V=\frac{1}{2}{m}^2{\varphi }^2$ this is the Klein-Gordon equation, and with $V=0$ it is the wave equation in four dimensions — the string's (6) with three space dimensions instead of one.

The symmetry chose the theory. Nothing was assumed about the dynamics; the requirement of Lorentz invariance plus locality plus first derivatives left almost no freedom. This is the characteristic method of modern physics and it is a direct consequence of taking the action rather than the equations of motion as the primitive object.

The stress-energy tensor

If $ℒ$ has no explicit dependence on $x$, the action is invariant under the four translations $x^{\alpha }\to x^{\alpha }+{\epsilon }^{\alpha }$. That is a four-parameter group, so Noether's first theorem gives four conserved currents. Assembled, they form the canonical stress-energy tensor

$T^{\alpha }{}_{\beta }=\frac{\partial ℒ}{\partial ({\partial }_{\alpha }{\varphi }_i)}{\partial }_{\beta }{\varphi }_i-{\delta }^{\alpha }{}_{\beta }ℒ\mathrm{(}\mathrm{5}\mathrm{)}$

satisfying ${\partial }_{\alpha }{T}^{\alpha }{}_{\beta }=0$.

Compare (5) with the Hamiltonian $H=p\dot{q}-L$ of the Hamiltonian and Hamilton's equations: the $\beta =0$ component is exactly that expression with the momentum replaced by the momentum density. The stress-energy tensor is the Legendre transform of the Lagrangian density, indexed by direction. Energy density, momentum density, energy flux and stress are its components.

Noether's own §6 concerns precisely these quantities. She calls the relations arising from the displacement group the energy relationships and their $B^{(\lambda )}$ the energy components, and asks when they are proper rather than improper — a question that becomes decisive on the Einstein-Hilbert action.

(5) is not the last word. The canonical tensor is generally neither symmetric nor gauge invariant, and it can be improved by adding a term whose divergence vanishes identically — which changes no conservation law. The symmetric improvement is the one that couples to gravity. Noether's classification anticipates this: adding an identically divergence-free quantity is exactly the freedom her improper divergence relations describe.

Fields with internal symmetry

For a complex scalar, $ℒ$ may be required invariant under

$\varphi \to e^{i\alpha }\varphi \mathrm{(}\mathrm{6}\mathrm{)}$

with $\alpha$ constant. This is a one-parameter group acting on the fields rather than on spacetime, and Noether gives one conserved current:

$j^{\alpha }=i({\varphi }^{\ast }{\partial }^{\alpha }\varphi -\varphi {\partial }^{\alpha }{\varphi }^{\ast })\mathrm{(}\mathrm{7}\mathrm{)}$

which is the electric charge current up to normalisation. Charge conservation is a symmetry of the action, and there is no other reason for it.

If $\alpha$ is allowed to depend on position, (6) becomes a group of arbitrary functions, and the theory falls under Noether's second theorem rather than the first. That change is the subject of the next page, and it forces the existence of a new field.

Provenance: every equation on this page is derived here or is standard. The characterisations of Noether §6 are paraphrased from the ESTABLISHED source under the translation rule stated on Noether's first theorem. See Sources.

Next

the electromagnetic Lagrangian makes the phase in (6) local.