From Newtonian Mechanics to Quantum Electrodynamics: A Unified Framework on the Invariance and Transformation of Mass and Energy

1. Introduction

The relationship between mass and energy lies at the core of physical principles spanning classical mechanics, special relativity [1,2,3,4,5], and quantum electrodynamics. In classical Newtonian physics, mass ($m$) is treated as an immutable, invariant scalar quantity that serves as a measure of inertia and gravitational attraction. Within this macroscopic regime, interactions occurring at velocities far below the speed of light ($v\ll c$) govern physical phenomena through mechanical work, kinetic energy, and potential energy transformations without altering system mass.

However, the advent of special relativity fundamentally redefined mass not as an absolute constant, but as a dense, bound form of intrinsic energy. Through Einstein’s formulation ($E=m{c}^2$), mass and energy were unified into a singular physical entity. In nuclear reactions, radioactive decay, and matter-antimatter annihilation, rest-frame mass can diminish or disappear entirely, re-emerging as radiant field energy. This transition is visualized in Figure 1, illustrating the complete conversion of a resting mass into quantized radiation.

2. Classical Mechanics and Mass Invariance

In classical Newtonian mechanics, macroscopic objects operating at non-relativistic velocities ($v\ll c$) undergo mechanical work, energy storage, and energy transfer without measurable variations in rest mass ($m$).

When an object of mass $m$ is launched into motion:

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Kinetic Energy ($\mathit{K}.\mathit{E}.$): Work performed by applied external forces converts into motion:

$$K.E.={\displaystyle \frac{1}{2}}m{v}^2$$

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Potential Energy ($\mathit{P}.\mathit{E}.$): Work performed against a conservative field (e.g., gravity) is stored as positional energy:

$$P.E.=mgh$$

Under these conditions, mass remains an invariant scalar quantity ($dm/dt=0$). Energy transforms between mechanical forms (kinetic to potential and vice versa) without mass deficit or mass creation. Consequently, invoking Einstein’s mass-energy equivalence equation ($E=m{c}^2$) in classical mechanics is unnecessary and physically inappropriate.

3. Special Relativity and Rest-Frame Energy

When particles undergo high-energy interactions, radioactive decay, or complete annihilation, mass can no longer be treated as an invariant constant separated from energy. Instead, mass itself represents a concentrated form of rest-frame energy.

For a stationary particle or system of mass $m$, its intrinsic rest energy $E_{rest}$ is given by Einstein’s mass-energy equivalence formula:

$$E_{rest}=m{c}^2$$

where $m$ is the rest mass (invariant mass) of the system and $c$ is the speed of light in vacuum ($c\approx 3\times {10}^8\mathrm{m}/\mathrm{s}$).

In processes where rest mass decreases by a deficit $∆m$ (such as nuclear fission or fusion), energy is liberated according to $∆E=∆m{c}^2$.

4. Quantum Electrodynamics: Planck-Einstein Formulation

When a resting mass $m$ transitions into radiation, the liberated energy manifests as quantized electromagnetic energy packets (photons). According to Planck’s quantum postulate, the energy of a photon is directly proportional to its frequency ($f$) or inversely proportional to its wavelength ($\lambda$):

$$E=hf={\displaystyle \frac{hc}{\lambda }}$$

where:

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$h$ is Planck’s constant ($6.626\times {10}^{-34}\mathrm{J}.\mathrm{s}$),

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$f$ is the radiation frequency,

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$\lambda$ is the radiation wavelength ($c=f\lambda$).

This formulation reflects the fundamental wave-particle duality of light, expressing energy directly through quantum frequency parameters without requiring particle mass terms. The final state, consisting of quantized photons carrying this energy, is a direct result of the Complete Radiative Transition process described in Figure 1.

5. Total Energy Conservation and Field Synthesis

When a resting mass $m$ undergoes complete transformation into radiation emitting multiple photons of varying or equal frequencies ($f_1,f_2,f_3.\cdots$), the total energy of the emitted photonic field is the summation of individual photon energies [2]:

$$E_{radiation}=\sum _{i}h{f}_i$$

Enforcing the Law of Conservation of Energy:

$${E_{initial}=E}_{radiation}$$

$$m{c}^2=\sum hf$$

This conservation is illustrated in the transition diagram below.

Special Case: Electron-Positron Annihilation

In electron-positron pair annihilation at rest, the total initial rest mass of the electron-positron system ($2m_e$) converts completely into gamma-ray photon energy ($e^{-}+e^{+}\to {\gamma }_1+{\gamma }_2$):

$$E_{total}=2m_e{c}^2=h{f}_1+h{f}_2=2hf\Rightarrow m_e{c}^2=hf$$