5.1 Why Do Photons Have No Mass? — Pure Translation Modes

In the Light Path Conservation Axiom $v_{ext}^2+v_{int}^2=c^2$, we have already seen the shadow of mass. Rest mass $m_0$ corresponds to the maximum internal oscillation frequency when $v_{ext}=0$.

For photons, experiments tell us their rest mass is zero ($m_{\gamma }<10^{-18}$ eV). This means in our theory, photons must satisfy:

$$v_{int}\equiv 0$$

which leads to:

$$v_{ext}\equiv c$$

Why must photons be this way? This stems from their evolution mode in QCA networks.

5.1.1 Definition 5.1 (Translation-Invariant Mode)

Consider a one-dimensional simplified model of QCA. Let quantum state be $|\psi (x,t)⟩$.

If evolution operator $\hat{U}$ merely maps state from $x$ to $x+a$ (or $x-a$) without changing its internal phase or spin structure, then this excitation is called a Pure Translation Mode.

$$\hat{U}|\psi (x)⟩=|\psi (x\pm a)⟩$$

5.1.2 Theorem 5.1 (Massless Theorem)

If an excitation’s evolution operator $\hat{U}$ can be globally diagonalized as pure translation generator $e^{i\hat{P}a}$ without any local mixing terms, then this excitation’s internal evolution speed $v_{int}=0$, i.e., this particle is massless.

Proof:

Pure translation means Hamiltonian $\hat{H}\sim \hat{P}$.

In Hilbert space, state vector $|\psi (t)⟩$ merely moves between position bases $|x⟩$ without evolving on internal degrees of freedom (such as spin flips).

According to microscopic derivation of Light Path Conservation (see Chapter 3, Section 3.2), $v_{int}$ corresponds to parts of Hamiltonian that anti-commute with momentum $\hat{P}$ (i.e., mass term $\hat{M}$).

If $\hat{H}$ only has $\hat{P}$ term, then $\hat{M}\equiv 0⟹v_{int}\equiv 0$.

Q.E.D.

5.1.3 Physical Picture

Photons are “heartless” information packets. When transmitting on lattices, they need no internal computation. They simply transport bits from A to B.

Because they have no “internal life,” they have no proper time ($\tau =0$) and no rest mass. They are destined to eternally wander at the universe’s maximum allowed speed.

This also explains why gauge bosons (under unbroken symmetry) are usually massless: they are messengers for transmitting information in networks, not nodes for processing information.

So, what has mass? It must be things that “stop to think.” In the next section, we will see that matter particles are precisely products of this “thinking.”