Quantizing the scalar field

Use $\hbar=c=1$, $\eta=\operatorname{diag}(1,-1,-1,-1)$, and the stable massive real scalar action from the preceding lesson. Its canonical momentum is $\pi=\dot\phi$. At equal time we impose a commutator between the field and momentum, while fields commute with fields and momenta commute with momenta. The spatial delta function plays the role of the Kronecker delta for a continuum of oscillator coordinates.

A field at a single point is more singular than an ordinary operator matrix: quantum fields are operator-valued distributions. Smearing with a smooth test function produces the meaningful integrated observables used in a careful formulation. In introductory calculations we write pointwise expressions as shorthand, but products at coincident points can diverge. That warning is already present before interactions and explains why a regulator is a tool with a purpose, rather than cosmetic algebra.

Choose $p^0=E_{\mathbf p}=\sqrt{\mathbf p^2+m^2}$ and $p\cdot x=E_{\mathbf p}t-\mathbf p\cdot\mathbf x$. The annihilation part carries $e^{-ip\cdot x}$ and the creation part its conjugate. The factor $1/\sqrt{2E_{\mathbf p}}$ works with our commutator and integration measure. Other books distribute these factors differently; an isolated factor has no meaning until the state normalization is also stated.

The vacuum is annihilated by every $\hat a_{\mathbf p}$. Applying creation operators builds Fock states, with unrestricted nonnegative occupation of each bosonic mode. Momentum eigenstates in infinite volume are delta-normalized rather than normalizable. A localized preparation is a wave packet built by integrating momentum modes with a square-integrable envelope. Particle number is well defined for this free theory, but need not be conserved once interactions are introduced.

A number state has a sharp occupation, but its field expectation is zero: the field changes occupation by one, making the relevant states orthogonal. Its fluctuations are nevertheless nonzero. A coherent state, a superposition of different occupations with a definite phase relationship, can have an expectation value following a classical free-field solution. This is the bridge to a classical wave, rather than equating a one-particle wavefunction with a directly measured classical field.

The free Hamiltonian sums oscillator energies, including a formally divergent zero-point contribution. Normal ordering subtracts the free vacuum constant when computing excitation energies in nongravitational flat-space calculations. This does not solve the cosmological constant problem or erase all physically relevant vacuum effects. What we can calculate here without ambiguity is the energy difference between finite-occupation states with the same regulator and reference vacuum.