The concept of a scalar in physics is essentially the same as in mathematics. Formally, a scalar is unchanged by coordinate system transformations. In classical theories, like Newtonian mechanics, this means that rotations or reflections preserve scalars, while in relativistic theories, Lorentz transformations or space-time translations preserve scalars.

Scalar field[edit]

Main articles: Scalar field and Scalar field theory

Since scalars mostly may be treated as special cases of tensors (as is done with vectors, ...), physical scalar fields are a special case of more general fields, like vector fields, spinor fields, and tensor fields.

Physical quantity[edit]

Main article: Physical quantity

A physical quantity is expressed as the product of a numerical value and a physical unit, not merely a number. The quantity does not depend on the unit (e.g. for distance, 1 km is the same as 1000 m), although the number depends on the unit. Thus, following the example of distance, the quantity does not depend on the length of the base vectors of the coordinate system. Also, other changes of the coordinate system may affect the formula for computing the scalar (for example, the Euclidean formula for distance in terms of coordinates relies on the basis being orthonormal), but not the scalar itself. In this sense, physical distance deviates from the definition of metric in not being just a real number; however it satisfies all other properties. The same applies for other physical quantities which are not dimensionless.