Other systems where particle number is not conserved involvephononsandspin waves in
condensed matter problems. Phonons are the quanta associated with vibrational modes of a crystal
or fluid, while spin waves are associated with fluctuating spins. The number of particles is also not
conserved in nuclear processes like fusion and fission.
18.1 Relativity and quantum mechanics
Special relativity invariably implies that the number of particles is not conserved. Indeed, one of
the key results of special relativity is the fact that mass isa form of energy. A particle at rest with
massmhas a rest energy given by the famous formula
E=mc^2 (18.1)The formula also implies that, given enough energy, one cancreate particlesout of just energy –
kinetic energy for example. This mechanism is at work in fire and light bulbs, where energy is
being provided from chemical combustion or electrical input to excite atoms which then emit light
in the form of photons. The mechanism is also being used in particle accelerators to produce new
particles through the collision of two incoming particles. In Figure??the example of a photon
scattering off an electron is illustrated. In (c), a photon oflow energy (≪mec^2 ) is being scattered
elastically which results simply in a deflection of the photon and a recoil of the electron. In (d), a
photon of high energy (≫mec^2 ) is being scattered inelastically, resulting not only in a deflection
of the photon and a recoil of the electron, but also in theproduction of new particles.
The particle data table also provides numerous examples of particles that are unstable and
decay. In each of these processes, the number of particles isnot conserved. To list just a few,
n → p++e−+ ̄νe
π^0 → γ+γ
π+ → μ++νμ
μ+ → e++νe+ ̄νμAs already mentioned, nuclear processes such as fusion and fission are further examples of systems
in which the number of particles is not conserved.
18.2 Why Quantum Field Theory?
Quantum Field Theory is a formulation of a quantum system in which the number of particles does
not have to be conserved but may vary freely. QFT does not require a change in the principles of
either quantum mechanics or relativity. QFT requires a different formulation of the dynamics of
the particles involved in the system.
Clearly, such a description must go well beyond the usual Schr ̈odinger equation, whose very
formulation requires that the number of particles in a system be fixed. Quantum field theory may
be formulated for non-relativistic systems in which the number of particles is not conserved, (recall