Particle Collisions

When two particles collide, extraordinary things can happen - they might bounce off each other, exchange energy, or even transform into brand-new particles.

In modern physics, a collision isn’t a literal crash like billiard balls colliding. It’s an interaction between fields, where energy and force are exchanged at the tiniest scales of nature.

At the subatomic level, particles never actually “touch.” Their interactions take place through the exchange of force-carrying bosons - photons for electromagnetic forces, gluons for the strong force, and W and Z bosons for the weak force.

Conservation Laws

In every particle collision, the fundamental conservation principles of relativistic physics hold true:

• Conservation of Total Energy
  The overall energy of the system remains the same before and after the collision.

  For instance, during a particle collision, the system’s total energy is conserved. In other words, the sum of the particles’ energies before the interaction equals the sum after it: $$ E_1 + E_2 = E_3 + E_4 $$ The outgoing particles (3 and 4) may be the same as those that initially collided (1 and 2), or entirely new particles created in the impact. The first case is known as an elastic collision (same particles), whereas the second describes an inelastic reaction (different particles), where part of the kinetic energy is transformed into other forms-such as rest mass or internal energy.

• Conservation of Momentum
  The total momentum vector of the system remains constant.

  Just like energy, the total momentum of the system is conserved during a collision. In vector form, this law is written as: $$ \vec{p}_1 + \vec{p}_2 = \vec{p}_3 + \vec{p}_4 $$ Here, $ p_1 $ and $ p_2 $ represent the momenta of the incoming particles, while $ p_3 $ and $ p_4 $ correspond to those of the outgoing ones. This relation shows that the vector sum of momenta before the impact is identical to that afterward-momentum is neither created nor destroyed, only redistributed among the collision products.

• Conservation of Electric Charge (and Other Quantum Numbers)
  The total electric charge, baryon number, lepton number, and other conserved quantities remain unchanged throughout the interaction.

Hence, all fundamental physical quantities-energy, momentum, and charge-are conserved in every interaction.

The only exception is kinetic energy $ K $, which may or may not be conserved depending on the type of interaction.

Note. In special relativity, the conservation of energy and momentum are unified into a single, elegant formulation expressed through the energy-momentum four-vector: $$ P_1 + P_2 = P_3 + P_4 $$ Each term ($P_i$) represents a four-vector of the form $$ P_i = \left( \frac{E_i}{c},\ \vec{p}_i \right) $$ combining a particle’s relativistic energy and momentum into one entity. This formulation is more general, as it holds in all inertial reference frames and provides a unified framework for describing both elastic collisions and particle creation or annihilation processes, where kinetic energy may be converted into mass energy or vice versa.

The Difference Between Classical and Relativistic Physics

In classical physics, the conservation laws take a different form from those in relativity.

During a collision between two bodies, the total momentum of the system is conserved, but the kinetic energy may not be-for example, part of it can be transformed into heat or deformation energy. The same principle appears in relativistic theory, though expressed differently.

However, classical mechanics always obeys the law of conservation of mass, written as:

$$ m_1 + m_2 = m_3 + m_4 $$

The total mass of the interacting bodies therefore remains constant before and after the collision. This means that no mass is ever created or destroyed; it can only shift or redistribute among the objects involved.

By contrast, in relativistic physics, mass is not conserved. It can be converted into energy (and vice versa) according to Einstein’s celebrated relation:

$$ E = mc^2 $$

This marks a fundamental distinction between the two frameworks: in classical physics, mass is conserved, whereas in relativistic physics, it is the total energy-including the energy associated with rest mass-that remains invariant.

In relativistic physics, we no longer speak of transformations between different forms of energy - mechanical, thermal, elastic, and so on - as we do in classical mechanics. Instead, energy is described in terms of two fundamental components: rest energy and kinetic energy. Together, these define the total energy state of any particle.

Types of Collisions

Collisions are generally grouped into three main categories:

• Elastic collisions
  In an elastic collision, the total kinetic energy of the system is conserved. The particles may change direction, but they do not transform into new ones - their rest masses remain the same.

  A classic example is the elastic scattering of an electron by a proton: $$ p + e \rightarrow p + e $$ In this process, both the proton and the electron emerge unchanged after the interaction. No new particles are created or destroyed, and the total kinetic energy remains constant (in the center-of-mass frame).

• Inelastic (or adhesive) collisions
  In an inelastic collision, part of the kinetic energy is converted into other forms of internal energy - such as excitation energy, heat, or mass energy - collectively referred to as rest energy.

  A well-known example is nuclear fusion, where two nuclei merge to form a heavier one. Part of the incoming nuclei’s kinetic energy is converted into the rest energy of the new, bound system.

• Explosive (or decay) collisions
  In an explosive collision, a single particle disintegrates into several products that carry away a greater total amount of kinetic energy. Here, a portion of the original particle’s rest energy is transformed into the kinetic energy of the fragments, in accordance with the mass - energy equivalence principle.

  For example, a neutron can decay into a proton, an electron, and an antineutrino: $$ n \rightarrow p + e^- + \bar{\nu}_e $$ In this decay, part of the neutron’s mass is released as the kinetic energy of the resulting particles.

Where Mass and Energy Meet

At very high energies, collisions reveal one of nature’s most fascinating truths: mass and energy are interchangeable.

After a collision, the total mass of the resulting particles might differ from the original, but the sum of energy and momentum always stays constant.

Part of the initial motion (kinetic energy) can convert into mass - creating entirely new particles from pure energy - and the reverse can happen too.

Example

In giant particle accelerators like CERN’s Large Hadron Collider, beams of protons or electrons are accelerated to nearly the speed of light and then smashed together.

The goal is to transform kinetic energy into the mass of new particles, following Einstein’s famous equation:

$$ E = mc^2 $$

Head-on collisions release the greatest amount of usable energy, which is why modern accelerators are designed to make particles collide directly, rather than firing them at a fixed target.

Real-world example. Two protons collide head-on at several tera-electronvolts (TeV). The enormous energy concentrated at the collision point can “condense” into new particles - W and Z bosons, or quark-antiquark pairs, for instance. The reaction can be represented as: $$  p + p \rightarrow p + p + X $$ where X stands for everything created in the interaction - new particles, radiation, and other energy forms.

Through such collisions, scientists explore the deepest structure of the universe - revealing how energy becomes matter, and how matter, in turn, can dissolve back into pure energy.