Quantum Scalar Field Theory
In quantum field theory, the fields, and all observables constructed from them, are replaced by quantum operators on a Hilbert space. This Hilbert space is built on a vacuum state, and dynamics are governed by a Hamiltonian, a positive operator which annihilates the vacuum. A construction of a quantum scalar field theory may be found in the canonical quantization article, which uses canonical commutation relations among the fields as a basis for the construction. In brief, the basic variables are the field φ and its canonical momentum π. Both fields are Hermitian. At spatial points at equal times, the canonical commutation relations are given by
and the free Hamiltonian is
A spatial Fourier transform leads to momentum space fields
which are used to define annihilation and creation operators
where . These operators satisfy the commutation relations
The state |0> annihilated by all of the operators a is identified as the bare vacuum, and a particle with momentum is created by applying to the vacuum. Applying all possible combinations of creation operators to the vacuum constructs the Hilbert space. This construction is called Fock space. The vacuum is annihilated by the Hamiltonian
where the zero-point energy has been removed by Wick ordering. (See canonical quantization.)
Interactions can be included by adding an interaction Hamiltonian. For a φ4 theory, this corresponds to adding a Wick ordered term g:φ4:/4! to the Hamiltonian, and integrating over x. Scattering amplitudes may be calculated from this Hamiltonian in the interaction picture. These are constructed in perturbation theory by means of the Dyson series, which gives the time-ordered products, or n-particle Green's functions as described in the Dyson series article. The Green's functions may also be obtained from a generating function that is constructed as a solution to the Schwinger-Dyson equation.
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