Physical quantity

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A physical quantity is either a quantity within physics that can be measured (e.g. mass, volume, etc.), or the result of a measurement. Physical quantities are usually expressed as the product of a numerical value and a physical unit (SI units are usually preferred today). The notion of physical dimension of a physical quantity was introduced by Fourier (1822).

Contents

1 Example
2 Symbols for physical quantities
3 Extensive and intensive quantities
4 Physical quantities as ''coordinates'' over spaces of physical ''qualities''
5 See also
6 Books

Example

If a certain value of power is written as

P = 42.3 x 10³ W = 42.3 kW,

then

P represents the physical quantity of power

42.3 x 10³ is the numerical value

k is the SI prefix kilo, representing 10³

W is the symbol for the unit of power, the watt

Symbols for physical quantities

Usually, the symbols for physical quantities are chosen to be a single lower case or capital letter of the Latin or Greek alphabet written in italic type. Often, the symbols are modified by subscripts and superscripts, in order to specify what they pertain to - for instance E_p is usually used to denote potential energy and c_p heat capacity at constant pressure.

Extensive and intensive quantities

A quantity is called:

extensive when its magnitude is additive for subsystems (volume, mass, etc.)
intensive when the magnitude is independent of the extent of the system (temperature, pressure, etc.)

Some extensive physical quantities may be prefixed in order to further qualify their meaning:

specific is added to refer to the quantity divided by its mass (such as specific volume)
molar is added to refer to the quantity divided by the amount of substance (such as molar volume)

There are also physical quantities that can be classified as neither extensive nor intensive, for example angular momentum, area, force, length, and time.

Physical quantities as coordinates over spaces of physical qualities

The meaning of the term physical quantity is generally well understood (everyone understands what it is meant by the frequency of a periodic phenomenon, or the resistance of an electric wire). It is clear that behind a set of quantities like temperature − inverse temperature − logarithmic temperature, there is a qualitative notion: the cold−hot quality. Over this one-dimensional quality space, we may choose different coordinates: the temperature, the inverse temperature, etc. Other quality spaces are multidimensional. For instance, to represent the properties of an ideal elastic medium we need 21 coefficients, that can be the 21 components of the elastic stiffness tensor [ c_ ] , or the 21 components of the elastic compliance tensor (inverse of the stiffness tensor), or the proper elements (six eigenvalues and 15 angles) of any of the two tensors, etc. Again, we are selecting coordinates over a 21-dimensional quality space. On this space, each point represents a particular elastic medium.

It is always possible to define the distance between two points of any quality space, and this distance is —inside a given theoretical context— uniquely defined. For instance, two periodic phenomena can be characterized by their periods, [ T_1 ] and [ T_2 ], or by their frequencies, [ \nu_1 ] and [ \nu_2 ] . The only definition of distance that respects some clearly defined invariances is [ D = | ]log[ (T_2/T_1 ) | = | ]log[ (\nu_2/\nu_1 ) | ] .

These notions have implications in physics. As soon as we accept that behind the usual physical quantities there are quality spaces, that usual quantities are only special coordinates over these quality spaces, and that there is a metric in each space, the following question arises: Can we do physics intrinsically, i.e., can we develop physics using directly the notion of physical quality, and of metric, and without using particular coordinates (i.e., without any particular choice of physical quantities)? In fact, physics can (and must?) be developed independently of any particular choice of coordinates over the quality spaces, i.e., independently of any particular choice of physical quantities to represent the measurable physical qualities. This point of view has recently been developed (Tarantola, 2006 [link] ).

Books

Cook, Alan H. The observational foundations of physics, Cambridge, 1994. ISBN 0521455979.
Fourier, Joseph. Théorie analytique de la chaleur, Firmin Didot, Paris, 1822. (In this book, Fourier introduces the concept of physical dimensions for the physical quantities.)
Tarantola, Albert. Elements for physics - Quantities, qualities and intrinsic theories, Springer, 2006. ISBN 3540253025. [link]

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