Scalar field theory

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.^[1]

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.^[2]

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.^[3]

The signature of the metric employed below is $(+, -, -, -)$ .

^ i.e., it transforms under the trivial $(0, 0)$ -representation of the Lorentz group, leaving the value of the field at any spacetime point unchanged, in contrast to a vector or tensor field, or more generally, spinor-tensors, whose components undergo a mix under Lorentz transformations. Since particle or field spin by definition is determined by the Lorentz representation under which it transforms, all scalar (and pseudoscalar) fields and particles have spin zero, and are as such bosonic by the spin statistics theorem. See Weinberg 1995, Chapter 5
^ This means it is not invariant under parity transformations which invert the spatial directions, distinguishing it from a true scalar, which is parity-invariant.See Weinberg 1998, Chapter 19
^ Brown, Lowell S. (1994). Quantum Field Theory. Cambridge University Press. ISBN 978-0-521-46946-3. Ch 3.

[1] .e., it transforms under the trivial $(0, 0)$ -representation of the Lorentz group, leaving the value of the field at any spacetime point unchanged, in contrast to a vector or tensor field, or more generally, spinor-tensors, whose components undergo a mix under Lorentz transformations. Since particle or field spin by definition is determined by the Lorentz representation under which it transforms, all scalar (and pseudoscalar) fields and particles have spin zero, and are as such bosonic by the spin statistics theorem. See Weinberg 1995, Chapter 5

[2] This means it is not invariant under parity transformations which invert the spatial directions, distinguishing it from a true scalar, which is parity-invariant.See Weinberg 1998, Chapter 19

[3] Brown, Lowell S. (1994). Quantum Field Theory. Cambridge University Press. ISBN 978-0-521-46946-3. Ch 3.

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