Quantum field theory

The Schrödinger equation of quantum mechanics is not Lorentz invariant. In order to obtain a Lorentz invariant equation of motion, special relativity must be incorporated in theory. The combination of quantum theory and special relativity results in relativistic quantum field theory.

In a quantum field theory a scalar, or pseudoscalar, Lagrangian is not sufficient to guarantee relativistic invariance, we must also make sure that the fields obey the operator requirements needed to certify Lorentz covariance. Imposing Lorentz covariance means that physical observables perceived in different Lorentz frames become related.

Quantization proceeds in the Heisenberg picture by interpreting the fields as operators on a Hilbert space, and requiring that the canonical commutations relations be satisfied. For a bosonic field $\phi(x)$, they are

$$[\pi({\bf {x}},t),\phi({\bf {x}},t)]= -i\delta^{3}({\bf {x}}-{\bf {y}})$$ (24)

${\hspace{49mm}}$ $[\phi({\bf {x}},t),\phi({\bf {y}},t)]$ = $[\pi({\bf {x}},t),\pi({\bf {y}},t)]$ = 0

Solutions to the equations of motion $i\dot{\phi}=[\phi,H]$, are given by the space of states and specification of how the field operators act on the states.

A field theory is required to have a ground state, otherwise all states are unstable against decays. In classical field theories the total energy is defined as the integral of the positive definite energy density $T_{00}$, in general relativityity the situation is however more subtle [23]. This positivity of energy is desired since it ensures the stability of the ground state. In quantum field theory the vacuum $\vert>$ is the lowest energy Poincaré invariant state where all real particles are absent. The states and the action of the field operators on them can be defined by means of $\vert>$ and the Green's functions

$$G_{N}(x_{1},...,x_{N})=<0\vert T(\phi(x_{1})...\phi(x_{n}))\vert>$$

where $T$ denotes time ordering.

The fields contain creation and annihilation operators, and all states can be constructed from the vacuum by repeated application of the particle creation operators. Taking interactions into account gives a vacuum consisting of virtual particle-antiparticle pairs which continually appear and disappear. And if enough energy is supplied, particles and antiparticles are created together out of the vacuum. This tells us not only how particles are created in high-energy accelerators, but also about the quantum processes that took place in the early universe, according to the hot Big Bang Model.
Because of the creation- and annihilation-processes, theory of interaction of relativistic particles is a many-body theory, to which only approximate, perturbative solutions are known.