Canonical field commutators and Hamiltonians

Retain natural units and the plus, minus, minus, minus metric. Expanding the Lorentz contraction gives a positive time-derivative square and negative spatial-gradient square in the Lagrangian. Differentiating with respect to the time derivative gives the canonical momentum. The Legendre transform reverses the spatial and mass contributions to produce a positive Hamiltonian density when mass squared is nonnegative. This sign check is essential before assigning a particle spectrum.

A boundary condition is part of the derivation. Integrations by parts assume periodic boundaries or sufficiently rapid decay, so no unaccounted surface work changes the conserved energy. A finite spatial lattice or truncated mode set gives a concrete quadratic system before continuum limits. Negative mass squared would change the stability of the chosen background rather than merely rename the oscillator frequency; the positive-mass free Fock construction used here would no longer describe that background in the same way.

Canonical field and momentum at the same time obey a commutator proportional to a spatial delta function, while field–field and momentum–momentum commutators vanish. This is not an instruction to assign an infinite value to a pointwise uncertainty. Smearing over space turns the delta into an overlap of test functions. Operators smeared over disjoint spatial supports have zero canonical overlap, and the degree of overlap determines the noncommutativity for intersecting probes.

At a regulator scale the delta kernel is represented by the identity on the retained modes or by a lattice Kronecker delta with its volume factor. Truncation generally changes exact localization: a finite Fourier sum cannot reproduce a compact spatial delta. The canonical relation should therefore be checked at the level appropriate to the regulator. Passing to the continuum means convergence of smeared expressions on a common domain, rather than substituting a divergent coincidence value into ordinary arithmetic.

The Heisenberg rule multiplies a commutator with the Hamiltonian by positive imaginary one. Using the canonical bracket makes the field derivative equal its momentum. Applying the rule again gives the spatial Laplacian minus mass squared times the field, after integration by parts. Combining the two equations recovers the Klein–Gordon equation and checks both the canonical normalization and the Hamiltonian's relative coefficients.

In free momentum modes the normal-ordered Hamiltonian counts quanta with positive energies. Creation raises energy by the corresponding positive frequency, and the vacuum is annihilated by every annihilation operator. Adding a constant to the Hamiltonian changes neither these commutators nor energy gaps. An interacting local polynomial may formally be added to the density, but coincident products then require renormalization and do not automatically define a self-adjoint continuum Hamiltonian on the free Fock space.