A quantum field is a fundamental physical field that exhibits quantum mechanical properties throughout spacetime.Wherever there seems to be emptiness, invisible life seethes. The quantum field is an ocean in which every particle is a ripple on its surface. Photons are light waves on the electromagnetic field, electrons are waves on the electron field.
Maxwell's classical field theory (1860s) became the forerunner. In the 1920s–1930s, Dirac, Heisenberg, and Pauli developed quantum electrodynamics. Emmy Noether in 1918 proved a theorem linking field symmetries and conservation laws. Satyendra Nath Bose and Albert Einstein created statistics for bosonic fields in 1924. Wolfgang Pauli formulated the exclusion principle for fermionic fields and the spin-statistics theorem.
The field is described by a Lagrangian, from which the equations of motion are derived via the Euler-Lagrange equations (for example, the Dirac equation for fermions). Quantization occurs by imposing canonical commutation or anticommutation relations: [φ(x), π(y)] = iħδ³(x-y) for bosons, {ψ(x), ψ†(y)} = δ³(x-y) for fermions. The creation and annihilation operators in the field expansion act on Fock space. Interactions are introduced through gauge invariance. At high energies, quantum fields exhibit renormalizability.
Even in absolute vacuum, pairs of particles and antiparticles are constantly born and disappear, like bubbles in soda.
| f | variable | frequency of the quantum field oscillation — frequency hertz |
| m | variable | mass of the particle associated with the field — mass kilogram |
| c | constant | universal constant relating space and time in relativity — speed of light = 299792458 metre per second |
| h | constant | fundamental constant relating energy to frequency in quantum mechanics — planck constant = 6.62607015e-34 joule second |
| Φ_0 | variable | Magnetic flux quantum — magnetic flux weber |
| h | constant | Planck constant, the fundamental quantum of action — planck constant = 6.62607015e-34 joule second |
| e | constant | Elementary charge, the magnitude of charge of a single electron — elementary charge = 1.602176634e-19 coulomb |


















