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Concept# Scalar field theory

Summary

In theoretical physics, scalar field theory can refer to a relativistically invariant classical or quantum theory of scalar fields. A scalar field is invariant under any Lorentz transformation.
The only fundamental scalar quantum field that has been observed in nature is the Higgs field. However, scalar quantum fields feature in the effective field theory descriptions of many physical phenomena. An example is the pion, which is actually a pseudoscalar.
Since they do not involve polarization complications, scalar fields are often the easiest to appreciate second quantization through. For this reason, scalar field theories are often used for purposes of introduction of novel concepts and techniques.
The signature of the metric employed below is (+, −, −, −).
A general reference for this section is Ramond, Pierre (2001-12-21). Field Theory: A Modern Primer (Second Edition). USA: Westview Press. , Ch 1.
The most basic scalar field theory is the linear theory. Through the Fourier decomposition of the fields, it represents the normal modes of an infinity of coupled oscillators where the continuum limit of the oscillator index i is now denoted by x. The action for the free relativistic scalar field theory is then
where is known as a Lagrangian density; d4−1x ≡ dx ⋅ dy ⋅ dz ≡ dx1 ⋅ dx2 ⋅ dx3 for the three spatial coordinates; δij is the Kronecker delta function; and ∂ρ = ∂/∂xρ for the ρ-th coordinate xρ.
This is an example of a quadratic action, since each of the terms is quadratic in the field, φ. The term proportional to m2 is sometimes known as a mass term, due to its subsequent interpretation, in the quantized version of this theory, in terms of particle mass.
The equation of motion for this theory is obtained by extremizing the action above. It takes the following form, linear in φ,
where ∇2 is the Laplace operator. This is the Klein–Gordon equation, with the interpretation as a classical field equation, rather than as a quantum-mechanical wave equation.

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