The Substance of the World
Fields and Their Symmetries
Abstract
Modern physics has taught us something deeply surprising:
the world is not fundamentally made of solid particles, but of fields.
Mass, energy, and even matter itself emerge from how these fields behave and how they respect certain symmetries of nature.
This book explains, in clear and non-technical language, how the famous equation E=mc2 arises, how energy becomes mass, why fields are the true carriers of both, and how all of this is confirmed by real experiments and measurements, not speculation.
As Einstein insisted: “Everything should be made as simple as possible, but not simpler.”
This text follows that principle using today’s experimental knowledge.
1. What We Mean by “Fields”
A field is something that exists everywhere in space and time and can carry energy.
Examples:
- The electromagnetic field (light, radio waves)
- The electron field (from which electrons arise)
- The quark and gluon fields (inside protons and neutrons)
- The gravitational field (spacetime itself)
In modern physics, particles are not fundamental objects.
They are localized excitations of fields, similar to waves on the surface of the ocean.
This is not philosophy it is how all high-precision experiments are described today.
2. Fields Carry Energy (Directly Measured)
Every physical field has an energy density: a measurable amount of energy stored at each point in space.
For example:
- Light can heat objects.
- Electromagnetic fields exert pressure (measured in solar sails).
- Gravitational waves transport energy (directly detected by LIGO).
These are observed facts, not assumptions.
3. From Maxwell to Relativity: Why Energy and Mass Are Linked
In the 19th century, James Clerk Maxwell showed that light is a wave of the electromagnetic field and that it always travels at the same speed in empty space.
This forced a radical conclusion:
- Space and time must adjust so that this speed is always the same.
- This leads to special relativity.
Relativity unifies energy and momentum into a single object.
4. The Key Relation Between Energy, Momentum, and Mass
Formula
E²=(pc)²+(mc²)²
Legend (symbols explained)
- E = energy
- p = momentum
- m = mass
- c = speed of light
This equation has been verified in countless particle accelerator experiments.
5. The Special Case: Rest Energy
If an object is at rest, its momentum is zero.
That leaves:E=mc²
Meaning
Mass is energy in its rest form.
This is not a definition it is a measured equivalence.
6. Experimental Proofs (Not Theory)
6.1 Mass Turning into Energy
- Electron + positron annihilation:
- Mass disappears
- Two photons appear
- Measured energy matches mc² exactly
6.2 Energy Turning into Mass
- Particle creation in accelerators:
- Energy is converted into massive particles
- Threshold energies match predicted masses
6.3 Nuclear Binding Energy
- Atomic nuclei weigh less than their components
- Missing mass appears as released energy
- Precision confirmed to better than one part in a billion
6.4 Heated Objects Are Heavier
- Adding heat adds energy
- Added energy increases mass (measured with modern precision instruments)
7. Fields as the Origin of Most Mass
A crucial modern discovery:
Most mass does not come from fundamental particle masses.
Example: the proton
- Quark masses contribute ~1%
- Field energy and interactions contribute ~99%
That mass comes from:
- Strong-force fields
- Motion and binding energy
- Field self-interactions
This is experimentally confirmed by lattice quantum chromodynamics and scattering data.
8. The Role of Symmetry
Fields obey deep symmetries:
- Symmetry under motion (relativity)
- Symmetry under rotations
- Symmetry under internal transformations (gauge symmetries)
These symmetries:
- Enforce conservation of energy and momentum
- Determine which fields are massless or massive
- Fix how energy becomes mass
Mass itself is a consequence of symmetry structure, not an independent substance.
9. Gravity Confirms the Picture
In general relativity, gravity is sourced by energy, not just mass.
Formula
Spacetime curvature=Energy and momentum of fields
This explains:
- Why light bends near stars
- Why radiation gravitates
- Why energy density shapes the universe
Again: observed, measured, confirmed.
10. The Modern Conclusion
From today’s experimental standpoint:
- Fields are fundamental
- Energy is real and measurable
- Mass is a form of energy
- Symmetries determine how fields behave
- Matter is structured field energy
Final Statement
The “substance of the world” is not matter, but fields.
Mass is not a basic ingredie nt, but an emergent property of energy held in stable field structures.
E=mc² is not a theory it is a measured identity of nature.
This is Einstein’s insight, sharpened and confirmed by a century of experiments.
By Jan Klein
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