note-math

Action of a scalar field

Kinetic energy part

\frac{1}{2} | grad ϕ |^{2}

\frac{1}{2} | \partial ϕ |^{2}

where $grad ϕ \leftrightarrow \partial ϕ$ by metric duality $(\partial ϕ) (𝑋) = 𝑔 (grad ϕ, 𝑋)$

Mass part $- \frac{1}{2} 𝑚^{2} | ϕ |^{2}$

Klein--Gordon-Lagrangian_(tag)

\int 𝑑 𝑥 (\frac{1}{2} | \partial ϕ |^{2} \mp \frac{1}{2} 𝑚^{2} | ϕ |^{2})

\int_{ℝ^{1, 3}} 𝑑 𝑥 (\frac{1}{2} (\partial ϕ^{*} \cdot \partial ϕ \mp 𝑚^{2} ϕ^{*} ϕ))

let δ diffeomorphism $Δ ϕ$ , let the differential of the action be zero

0 = Δ 𝑆 = \int_{ℝ^{1, 3}} 𝑑 𝑥 Re (\partial Δ ϕ^{*} \cdot \partial ϕ \mp 𝑚^{2} Δ ϕ^{*} ϕ)

product rule $\partial^{†} (Δ ϕ^{*} \partial ϕ) = \partial Δ ϕ^{*} \cdot \partial ϕ + Δ ϕ^{*} \partial^{†} \partial ϕ$

In coordinates $\partial^{†} \partial ϕ = 𝑔^{𝜇 𝜈} \partial_{𝜇} \partial_{𝜈} ϕ$

$ℝ^{1, 3}$ δ diffeomorphism of field $Δ ϕ$ , is zero at the boundary (boundary of $ℝ^{1, 3}$ i.e. infinity) such that

\begin{matrix} \int_{ℝ^{1, 3}} 𝑑 𝑥 Re (\partial^{†} (Δ ϕ^{*} \partial ϕ)) & = & lim_{𝑟 \to \infty} \int_{ℚ^{1, 3} (\pm 𝑟)} (Δ ϕ^{*} \partial ϕ) \cdot 𝑛 \\ = & 0 \end{matrix}

Differential of the action

0 = - \int_{ℝ^{1, 3}} 𝑑 𝑥 Re Δ ϕ^{*} (\partial^{†} \partial ϕ \pm 𝑚^{2} ϕ)

for all $Δ ϕ$ , thus giving the Lagrange-equation, here called Klein--Gordon-equation_(tag)

\partial^{†} \partial ϕ \pm 𝑚^{2} ϕ = 0

let $∆ = div grad = \partial^{†} \partial = \partial \partial^{†}$

massless
$∆ ϕ = 0$
massive
$(∆ \pm 𝑚^{2}) ϕ = 0$
mass term => $- \frac{1}{2} 𝑉 (| ϕ |^{2})$
$(∆ + 𝑉^{'}) ϕ = 0$

The action uses the quadratic form $| grad ϕ |^{2}$ and the metric volume form $𝑑 Vol$

in $ℂ$ , $| grad ϕ |^{2} = {(grad ϕ)}^{†} grad ϕ$

Repeat the above steps for a general scalar field action

\int_{ℝ^{1, 3}} 𝑑 𝑥 𝐿 (ϕ, \partial_{𝑥} ϕ)

0 = Δ 𝑆 = \int_{ℝ^{1, 3}} 𝑑 𝑥 (\frac{\partial 𝐿}{\partial ϕ} \cdot Δ ϕ + \frac{\partial 𝐿}{\partial (\partial_{𝑥} ϕ)} \cdot \partial_{𝑥} Δ ϕ)

In coordinates $\frac{\partial 𝐿}{\partial (\partial_{𝑥} ϕ)} \cdot \partial_{𝑥} Δ ϕ = 𝑔^{𝜇 𝜈} \frac{\partial 𝐿}{\partial (\partial_{𝜇} ϕ)} \cdot \partial_{𝜈} Δ ϕ$

product rule

\partial_{𝑥}^{†} (\frac{\partial 𝐿}{\partial (\partial_{𝑥} ϕ)} \cdot Δ ϕ) = (\partial_{𝑥}^{†} \frac{\partial 𝐿}{\partial (\partial_{𝑥} ϕ)}) \cdot Δ ϕ + \frac{\partial 𝐿}{\partial (\partial_{𝑥} ϕ)} \cdot \partial_{𝑥} Δ ϕ

In coordinates $\frac{\partial 𝐿}{\partial (\partial_{𝑥} ϕ)} \cdot Δ ϕ = {(\frac{\partial 𝐿}{\partial (\partial_{𝜇} ϕ)} \cdot Δ ϕ)}_{𝜇 = 0, \dots, 3}$

Divergence + Stokes' theorem + zero boundary + forall $Δ ϕ$ , collecting $Δ ϕ$ , terms gives the Lagrange-equation

\frac{\partial 𝐿}{\partial ϕ} - \partial_{𝑥}^{†} \frac{\partial 𝐿}{\partial (\partial_{𝑥} ϕ)} = 0

Note that $ℝ$ valued fields are not compatible with $U (1, ℂ)$ 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 $SO (1, 3)$ invariant

Question motivation-of-plane-wave-solution_(tag)

Motivation for plane waves? Inspired by the appearance of $exp$ in the solutions of linear ODEs with constant coefficients, especially the harmonic oscillator eq $\frac{𝑑^{2} 𝑥}{𝑑 𝑡^{2}} \pm 𝜔^{2} 𝑥 = 0$ , similar to the first-order linearization of #link(<harmonic-oscillator>)[harmonic oscillator] $(\begin{matrix} \frac{𝑑}{𝑑 𝑡} \\ \frac{𝑑}{𝑑 𝑡} \end{matrix}) (\begin{matrix} 𝑥 \\ 𝑣 \end{matrix}) = (\begin{matrix} 1 \\ \mp 𝜔^{2} \end{matrix}) (\begin{matrix} 𝑥 \\ 𝑣 \end{matrix})$ , for the KG equation

(\begin{matrix} \partial \\ \partial^{†} \end{matrix}) (\begin{matrix} ϕ \\ 𝜓 \end{matrix}) = (\begin{matrix} - 𝟙 \\ \mp 𝑚^{2} \end{matrix}) (\begin{matrix} ϕ \\ 𝜓 \end{matrix})

Perform #link(<exponential-of-vector-field>)[$exp$ transformation] or #link(<integral-curve>)[integral curve]

Trigonometric case

exp 𝑥 (\begin{matrix} - 𝟙 \\ \mp 𝑚^{2} \end{matrix}) = (\begin{matrix} cos 𝑝 𝑥 & \frac{𝑝}{| 𝑝 |^{2}} sin 𝑝 𝑥 𝟙 \\ - 𝑝 sin 𝑝 𝑥 𝟙 & cos 𝑝 𝑥 \end{matrix})

where $| 𝑝 |^{2} = 𝑚^{2}$ , $\frac{𝑝}{| 𝑝 |^{2}}$ represents quadratic form inversion, and acts on $𝜓$ via inner product

Thus

ϕ_{𝑝} (𝑥) = ϕ (0) cos 𝑝 𝑥 + \frac{𝑝 \cdot 𝜓 (0)}{𝑚^{2}} sin 𝑝 𝑥

Or written in the form of a complex exponential

\begin{matrix} ϕ (𝑥) & = & \frac{1}{2} (ϕ (0) - i \frac{𝑝 \cdot 𝜓 (0)}{𝑚^{2}}) 𝑒^{i 𝑝 𝑥} + \frac{1}{2} (ϕ (0) + i \frac{𝑝 \cdot 𝜓 (0)}{𝑚^{2}}) 𝑒^{- i 𝑝 𝑥} \\ ≕ & 𝑎 (𝑝, i) 𝑒^{i 𝑝 𝑥} + 𝑎 (𝑝, - i) 𝑒^{- i 𝑝 𝑥} \end{matrix}

Hyperbolic case is similar

linear-superposition-of-KG-eq_(tag)

Linear superposition of plane waves also satisfies scalar field eq

$𝑚 \neq 0$

Integrate superposition on the hyperboloid ${𝑝^{2} = \pm 𝑚^{2}} = ℚ (1, 3) (\pm 𝑚^{2})$

metric & volume form come from the restriction of $ℝ^{1, 3}$

In the case of $+ 𝑚^{2}$ , it can be done on one sheet of the three-dimensional spacelike two-sheet hyperboloid ${ℍ 𝕪}^{3} = {𝑝^{2} = 𝑚^{2}, 𝑝_{0} > 0}$ , because the other sheet can be obtained by collecting coefficients $𝑎 (𝑝, i) + 𝑎 (- 𝑝, - i)$ , which is equivalent to a single sheet

ϕ (𝑥) = \int_{\begin{matrix} {ℍ 𝕪}^{3} \\ 𝕊 (Im ℂ) \end{matrix}} 𝑑 𝑝 𝑑 i (𝑎 (𝑝, i) 𝑒^{𝑝 𝑥 i})

${\pm i} = 𝕊 (Im ℂ) = 𝕊^{0}$ . For $ℍ, 𝕆$ , plane waves probably need to consider all unit imaginary numbers, so do we need to integrate over $𝕊 (Im ℍ) = 𝕊^{2}, 𝕊 (Im 𝕆) = 𝕊^{6}$ ?

For $ℝ$ value fields, $𝑎 (𝑝, - i) = {𝑎 (𝑝, i)}^{*}$ , and written as ${𝑎 (𝑝)}^{*}$

Add square-integrable condition to $𝑎 (𝑝, i)$ (integral on $𝑝^{2} = 𝑚^{2}$ ), and in order to make some derivatives of $ϕ$ also square-integrable (Sobolev) e.g. $\partial_{𝜇} ϕ$ , usually some "polynomial multiplication" square-integrable conditions are added to $𝑎 (𝑝, i)$ e.g. $𝑝_{𝜇} i 𝑎 (𝑝, i)$

On simple "projection to $ℝ^{3}$ coordinates" (not $SO (1, 3)$ invariant), using notation $(𝑝_{0}, 𝑝) \in ℝ^{1, 3}$

ϕ (𝑥_{0}, 𝑥) = \int_{ℝ^{3}} 𝑑 𝑝 \frac{1}{\sqrt{2 𝑝_{0}}} (𝑎 (𝑝, i) 𝑒^{𝑝_{0} 𝑥_{0} i} 𝑒^{- 𝑝 𝑥 i} + 𝑎 (𝑝, - i) 𝑒^{- 𝑝_{0} 𝑥_{0} i} 𝑒^{𝑝 𝑥 i})

with $𝑝_{0} = 𝑝_{0} (𝑝) = \sqrt{𝑚^{2} + 𝑝^{2}} > 0, 𝑎 (𝑝, i) = 𝑎 (𝑝_{0}, 𝑝, i)$

$𝑝^{2} = - 𝑚^{2}$ cannot be simply "projected" to $ℝ^{1, 2}$ , it's not a bijection in the first place

$𝑚 = 0$

Cannot directly use submanifold metric volume form because the metric is zero. Can we use the limit

𝑚 \to 0

? Use some limit of

𝑝_{0} (𝑚), 𝑝 (𝑚), 𝑎 (𝑚)

unitary-representation-KG-field_(tag)

For $𝐿^{2}$ superposition of free fields, there is an $𝐿^{2}$ inner product, and it is $SO (1, 3) ⋊ ℝ^{1, 3}$ invariant. Preserving quadratic form implies preserving inner product

Translation $𝑥 \to 𝑥 + 𝑎$ makes $| 𝑎 (𝑝) exp (𝑝 𝑎 i) |^{2} = | 𝑎 (𝑝) |^{2}$

Rotation $(𝑝 \to Λ 𝑝) \in SO (1, 3)$ is an isometry of $ℚ^{1, 3} (\pm 𝑚^{2})$ , which does not change the integral

\int_{ℚ^{1, 3} (\pm 𝑚^{2})} 𝑑 Vol | 𝑎 (Λ 𝑝) |^{2} = \int_{ℚ^{1, 3} (\pm 𝑚^{2})} 𝑑 Vol | 𝑎 (𝑝) |^{2}

This is called the unitary representation of the Poincare group $SO (1, 3) ⋊ ℝ^{1, 3}$ , spin 0 part, $\pm 𝑚^{2} \in ℝ$

try-to-define-plane-wave-in-metric-manifold_(tag)

Can the $exp$ of $(\begin{matrix} \partial \\ \partial^{†} \end{matrix}) (\begin{matrix} ϕ \\ 𝜓 \end{matrix}) = (\begin{matrix} - 𝟙 \\ \mp 𝑚^{2} \end{matrix}) (\begin{matrix} ϕ \\ 𝜓 \end{matrix})$ be generalized on a manifold? Note that this is a coordinate-free notation. If coordinates are used, it's not a constant coefficient PDE. Whether it's constant coefficient or not, one can try to exponentiate it.

Can it be generalized to symmetric spaces $ℚ^{1, 4} (- 1), ℚ^{2, 3} (1)$ ?

Does (δ) isometry preserve $𝐿^{2}$ superposition?

To construct particle-like wave packets, first find static solutions, then boost

Does $ℝ^{1, 1}$ spacetime, $ℝ$ valued scalar field with potential $𝑉 (ϕ) = ϕ^{4}$ or $sin ϕ$ provide possible multiparticle wave packet models? (Soliton type)

Question Infinite difficulties

Free field $| 𝑝, \pm i ⟩ = exp (\pm 𝑝 𝑥 i)$ 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 $ϕ$ around $| 𝑝 ⟩$ with $𝐿 (ϕ, \partial ϕ) - 𝐿 (| 𝑝, \pm i ⟩, \partial | 𝑝, \pm i ⟩)$ being integrable, and the derivative of the action at $| 𝑝 ⟩$ is zero

Another method is to first integrate in a finite region, and then take the limit to an infinite region