TFP: Perturbations Around the Vacuum

Perturbations Around the Vacuum

Start from the vacuum solution:

• $A(x) = A_0$ (constant amplitude)

• $\phi(x) = \omega t$ (background phase)

• $a_\mu(x) = -\omega \delta^0_\mu$ (cancels time evolution)

• $g_{\mu\nu} = \eta_{\mu\nu}$ (flat spacetime)

Now introduce small perturbations:

• $\phi(x) = \omega t + \delta\phi(x)$
  

• $a_\mu(x) = -\omega \delta^0_\mu + \delta a_\mu(x)$
  

These combine in the covariant derivative as:

$$D_\mu \phi = \partial_\mu \phi + a_\mu = \delta_\mu^0 \omega + \partial_\mu \delta\phi + \delta a_\mu - \omega \delta_\mu^0 = \partial_\mu \delta\phi + \delta a_\mu$$

So the entire dynamics will be governed by the perturbations:

$$\chi_\mu(x) \equiv \partial_\mu \delta\phi + \delta a_\mu$$

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Equation of Motion for $\delta a_\mu$

Start from the full equation:

$$\partial_\nu f^{\nu\mu} + A^2 (\partial^\mu \phi + a^\mu) = 0$$

Linearize:

• $A = A_0 + \delta A \Rightarrow A^2 \approx A_0^2$

• $f^{\mu\nu} = \partial^\mu \delta a^\nu - \partial^\nu \delta a^\mu$
  

• $\partial^\mu \phi + a^\mu = \chi^\mu$
  

Then to first order:

$$\partial_\nu (\partial^\nu \delta a^\mu - \partial^\mu \delta a^\nu) + A_0^2 \chi^\mu = 0 \Rightarrow \Box \delta a^\mu - \partial^\mu (\partial_\nu \delta a^\nu) + A_0^2 \chi^\mu = 0$$

Now substitute $\chi^\mu = \partial^\mu \delta\phi + \delta a^\mu$:

$$\Box \delta a^\mu - \partial^\mu (\partial_\nu \delta a^\nu) + A_0^2 (\partial^\mu \delta\phi + \delta a^\mu) = 0$$

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🔄 Coupled Equation for $\delta\phi$

From the perturbed version of:

$$\partial_\mu \left( A^2 (\partial^\mu \phi + a^\mu) \right) = 0 \Rightarrow A_0^2 \partial_\mu \chi^\mu = 0$$

So we have:

$$\partial_\mu (\partial^\mu \delta\phi + \delta a^\mu) = 0 \Rightarrow \Box \delta\phi + \partial_\mu \delta a^\mu = 0$$

This gives the continuity-like constraint between $\delta\phi$ and $\delta a^\mu$.

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🧠 Interpretation

Define:

• $\delta a^\mu \equiv \text{vector potential perturbation (gauge field)}$
  

• $\delta\phi \equiv \text{temporal phase fluctuation}$
  

Then:

Main Field Equation:

$$\Box \delta a^\mu + A_0^2 \delta a^\mu = \partial^\mu \left( \partial_\nu \delta a^\nu - A_0^2 \delta\phi \right)$$

This closely resembles the Proca equation, with a mass term $m^2 = A_0^2$, unless we fix gauge.

Constraint Equation:

$$\Box \delta\phi + \partial_\mu \delta a^\mu = 0 \Rightarrow \text{divergence of vector potential tied to scalar fluctuation}$$

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✅ Choose Gauge: Lorenz-type

Let’s define a gauge-fixing condition that simplifies these:

$$\partial_\mu \delta a^\mu = -A_0^2 \delta\phi$$

Then the main field equation becomes:

$$\Box \delta a^\mu + A_0^2 \delta a^\mu = 0$$

This is a massive wave equation:

• If $A_0 = 0$: recovers Maxwell’s equation in Lorenz gauge

• If $A_0 \ne 0$: gives Proca equation → photon acquires mass from background flow

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✅ Result: Maxwell’s Equations Emerge in Limit $A_0 \to 0$

In this limit:

• Background flow vanishes

• Gauge field dynamics becomes:

$$\Box \delta a^\mu = 0, \quad \partial_\mu \delta a^\mu = 0$$

Which is exactly Maxwell’s equations in Lorenz gauge.

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🔚 Summary

Aspect	Result
Perturbed gauge field	$\delta a^\mu$ satisfies wave equation with mass term
Emergent Maxwell equations	In limit $A_0 \to 0$, mass vanishes → Maxwell's theory
Scalar fluctuation $\delta\phi$	Enforces divergence constraint on $\delta a^\mu$
Gauge structure	Emerges naturally from flow coupling and symmetry