note-math

Spacetime translation does not fix the "boundary". The variation is non-zero. Similar to the case of time translation for a point particle

Generally, region change is given by by δ diffeomorphism

Using the change of variable formula

Swap differentiation and integration

Apply it to

Consider the derivative of the action for a variation that is a translation in the direction. let , first order derivative

use product rule

use Lagrange-equation

use divergence + zero boundary, to get

[energy-momentum-tensor-KG]

get

for all as coordinate-frame, so

for calculate

or

It can be seen that this EM tensor, after the index is lowered, is symmetric, so

In addition, the KG action is a real value, so its EM tensor is a real value

Assume that the EM tensor of can be integrated over . Use the notation . energy

General potential =>

The energy of a relativistic scalar field is real and positive

[conserved-spatial-integral-energy-KG]

Other components, e.g. , are invariant along the direction. Use , the integral of and its . The limit of region approximation is for hyperbolic geodesic spheres (multi-radius)

Example For plane wave expansion

Energy is (Question)

For fields, whether spatial rotation or boost, even if the Lagrangian is invariant, the action still changes

Now use the notation

• Spatial rotation of . If , then , so the tangent vector is

  Let be the spatial rotation axis, then the tangent vector will be

• Boost of . If , then , so the tangent vector is

  Let be the spatial boost axis, then the tangent vector will be (The spacetime metric has negative definite space)

Now use the notation . The tangent vectors for rotation and boost are collectively denoted as , acting as δ spacetime rotation on the field , as a δ diffeomorphism

Using the KG equation, rearranging terms, we get the zero-divergence conserved current

[angular-momentum-KG] let be the energy-momentum tensor of the KG field. The angular momentum of the field

or

let be the solution of KG eq. The phase change and its δ change belong to variations near the solution with fixed boundaries, so

Using product rule + divergence + Stokes' theorem + zero boundary

for all valued function , therefore

is called the 4-current of the KG field [current-gauge-KG]

Fixing the coordinates, and considering the 4-current components as quantities integrable over , then the zero component, i.e., charge conservation, holds [conserved-spatial-integral-charge-KG]

note that it's non positive defnite, is anti-Hermitian