Action of a scalar field
Kinetic energy part
or
where by metric duality
Mass part
[Klein--Gordon-Lagrangian]
or
let ฮด diffeomorphism , let the differential of the action be zero
product rule
In coordinates
ฮด diffeomorphism of field , is zero at the boundary (boundary of i.e. infinity) such that
Differential of the action
for all , thus giving the Lagrange-equation, here called [Klein--Gordon-equation]
let
-
massless
-
massive
-
mass term =>
The action uses the quadratic form and the metric volume form
in ,
Repeat the above steps for a general scalar field action
In coordinates
product rule
In coordinates
Divergence + Stokesโ theorem + zero boundary + forall , collecting , terms gives the Lagrange-equation
Note that valued fields are not compatible with gauge
Plane wave
- Period
- Wavelength
- 4-Wave number
-
Wave speed
- Massless ==> Wave speed = Speed of light
- With mass ==> wave speed < speed of light. And wave speed is not invariant
Question [motivation-of-plane-wave-solution]
Motivation for plane waves? Inspired by the appearance of in the solutions of linear ODEs with constant coefficients, especially the harmonic oscillator eq , similar to the first-order linearization of harmonic oscillator , for the KG equation
Perform transformation or integral curve
Trigonometric case
where , represents quadratic form inversion, and acts on via inner product
Thus
Or written in the form of a complex exponential
Hyperbolic case is similar
[linear-superposition-of-KG-eq]
Linear superposition of plane waves also satisfies scalar field eq
Integrate superposition on the hyperboloid
metric & volume form come from the restriction of
In the case of , it can be done on one sheet of the three-dimensional spacelike two-sheet hyperboloid , because the other sheet can be obtained by collecting coefficients , which is equivalent to a single sheet
. For , plane waves probably need to consider all unit imaginary numbers, so do we need to integrate over ?
For value fields, , and written as
Add square-integrable condition to (integral on ), and in order to make some derivatives of also square-integrable (Sobolev) e.g. , usually some โpolynomial multiplicationโ square-integrable conditions are added to e.g.
On simple โprojection to coordinatesโ (not invariant), using notation
with
cannot be simply โprojectedโ to , itโs not a bijection in the first place
[unitary-representation-KG-field]
For superposition of free fields, there is an inner product, and it is invariant. Preserving quadratic form implies preserving inner product
Translation makes
Rotation is an isometry of , which does not change the integral
This is called the unitary representation of the Poincare group , spin 0 part,
[try-to-define-plane-wave-in-metric-manifold]
Can the of be generalized on a manifold? Note that this is a coordinate-free notation. If coordinates are used, itโs not a constant coefficient PDE. Whether itโs constant coefficient or not, one can try to exponentiate it.
Can it be generalized to symmetric spaces ?
Does (ฮด) isometry preserve superposition?
To construct particle-like wave packets, first find static solutions, then boost
Does spacetime, valued scalar field with potential or provide possible multiparticle wave packet models? (Soliton type)
Question Infinite difficulties
Free field is not integrable, so it cannot be substituted into the Lagrangian and then integrated
One possibly less satisfying approach is to only consider the integrability of the difference. Consider around with being integrable, and the derivative of the action at is zero
Another method is to first integrate in a finite region, and then take the limit to an infinite region