What is an expectation value?

An ensemble ascribes to every sufficiently well-behaved quantity f an expectation value <f>. To justify the name 'expectation value', the map f—><f> must be linear and monotonic,

<α f + β g> = α <f> + β <g> for α, β in C, <f> ≥ <g> if f ≥ g

and satisfy a continuity condition.

The standard example, which gives this concept its name, is the mean value of a collection of instances of a random variable.

But as always in mathematics, the name giving example rarely covers the whole spectrum of usages. One must therefore not imagine an abstract ensemble as a concrete ensemble made up of many identical objects, but only see this as a sometimes useful, sometimes misleading picture.

In particular, the universe is described in the thermal interpretation by an ensemble, although the universe is unique, and there only exists one (At least this is the only closed physical system of which we can ever have knowledge! Because knowledge assumes interaction, and thus non-closure.)

Expectation values appear in many guises. The mathematical formalism describes not only the statistical evaluation of sequences of trials, but, for example, also statistical mechanics.

When I measure the temperature of a cup of tea, I make a single measurement and obtain a number, from which I can calculate the internal energy using the methods of thermodynamics and known properties of the substance 'tea'. I have involuntarily measured these at the same time. According to statistical mechanics, however, the internal energy is the expectation value of the microscopic energy operator. This expectation value satisfies all the mathematical requirements, although it is a single measurement of a single system - one does not need to measure hundreds of cups of tea or make hundreds of measurements.

The grand canonical ensemble observed here is purely fictional, introduced only to enable us to work with statistical concepts. It is just as fictional as the 6N-dimensional phase space, which we treat in analogy with 3 dimensional 'space' in order to give ourselves an intuitive picture.

Mathematical concepts are abstract by nature: to make use of them, it is not necessary that the actual significance of the concept is the same as that of the context in which the concept was originally developed. It is only necessary for the same formal conditions to be fulfilled.

Arnold Neumaier (Arnold.Neumaier@univie.ac.at)
A theoretical physics FAQ