manifest. Thus it is necessary to check that relativistic invariance is not lost. Alternatively, the Feynman integral approach is available for quantizing relativistic fields, and is manifestly invariant. For non-relativistic field theories, such as those used in condensed matter physics, Lorentz invariance is not an issue. Field operators Quantum mechanically, the variables of a field (such as the field's amplitude at a given point) are represented by operators on a Hilbert space. In general, all observables are constructed as operators on the Hilbert space, and the time-evolution of the operators is governed by the Hamiltonian, which must be a positive operator. A state annihilated by the Hamiltonian must be identified as the vacuum state, which is the basis for building all other states. In a non-interacting (free) field theory, the vacuum is normally identified as a state containing zero particles. In a theory with interacting particles, identifying the vacuum is more subtle, due to vacuum polarization, which implies that the physical vacuum in quantum field theory is never really empty. For further elaboration, see the articles on the quantum mechanical vacuum and the vacuum of quantum chromodynamics. The details of the canonical quantization depend on the field being quantized, and whether it is free or interacting. Real scalar field A scalar field theory provides a good example of the canonical quantization procedure. Classically, a scalar field is a collection of an infinity of oscillator normal modes. It suffices to consider a 1+1-dimensional space-time in which the spatial direction is compactified to a circle of circumference 2, rendering the momenta discrete. The classical Lagrangian density describes an infinity of coupled harmonic oscillators, labelled by which is now a label (and not the displacement dynamical variable to be quantized), denoted by the classical field , where is a potential term, often taken to be a polynomial or monomial of degree 3 or higher. The action functional is . The canonical momentum obtained via the Legendre transformation using the action is , and the classical Hamiltonian is found to be Canonical quantization treats the variables and as operators with canonical commutation relations at time = 0, given by Operators constructed from and can then formally be defined at other times via the time-evolution generated by the Hamiltonian, However, since and no longer commute, this expression is ambiguous at the quantum level. The problem is to construct a representation of the relevant operators on