A succinct presentation of the quantized Klein-Gordon field, and a similar quantum presentation of the classical Klein-Gordon random field

A succinct presentation of the algebraic structure of the quantized Klein-Gordon field can be given in terms of a Lorentz invariant inner product. A presentation of a classical Klein-Gordon random field at non-zero temperature can be given in the same non-commutative algebraic style, allowing a detailed comparison of the quantized Klein-Gordon field with a classical Klein-Gordon random field.