Two kinds of expectation values

The ambiguous meaning of the concept 'expectation value' is the cause of much misunderstanding. From a mathematical point of view, the expectation value is no more and no less than the image of a linear operator under a monotonic linear map. How we interpret this monotonic linear map is (as with any mathematical concept) up to us.

The traditional interpretation relates the concept to the fact that the average of a sequence of measurement values in similar ('identically prepared') scenarios ωk (k=1 to N)

<f>empirical = 1/N ∑k f(ωk)      (*)

has the properties of the mathematical expectation value (hence the name).

However the identification of this empirical concept with the mathematical one already causes philosophical difficulties, for example in the case of a Gaussian-distributed random variable, because one cannot say precisely how the formal variable is related to the measurements. One mumbles something about probability as the limiting value of relative frequencies (but only in the limiting case of infinitely many measurements, and only with probability 1) and has difficulties in ascribing a continuous distribution to a finite ensemble. Or one speaks of an imagined ensemble of all possible values that the measurement could have taken, in order to rationalise this. Both are signs that something sloppy is going on. Even at the classical level! The literature concerning the foundations of probability is for this reason just as extensive and controversial as that of quantum mechanics.

If one looks at a classical chaotic system, one has on the one hand deterministic dynamics, on the other hand behaviour which is in practice stochastic. This effectively stochastic behaviour can be seen in that with (*) at randomly selected times (which play the role of the scenarios ωk) one gets somewhat robust reproducible results, when f is a sufficiently nice observable. (And one can construct arbitrarily nasty observables…) The statistical averages arrived at in this way are typically, up to an error of order O(N{-1/2}), the same as the expectation values defined by

<f> = ∫ dμ f,

where μ is the invariant mass belonging to the attractor of the deterministic orbit. This mass, and thus all <f> , are objective properties of the system, just as much as the deterministic trajectory itself. These expectation values <f> , which are objectively determined by the invariant mass for single systems, satisfy the same rules as the empirical expectation value, in which one constructs average values from an ensemble of random-time samples. The <f> are therefore just as much objective observables of the singe system as the f themselves, only they are constant in time.

They can, however, only be measured approximately, just like the trajectory itself. But in contrast with the values of the trajectory, the <f> are approximately measurable in a reproducible sense!

With some modifications, which I do not wish to express more precisely here, the same thing holds in the instationary case, and one gets time-dependant objective expectation values. Of course these 'renormalised' observables <x(t)> are not the same as the 'bare' observables x(t), but both are objective properties of the system; only the prescriptions for measurement are different, since they correspond to different observables.

If one considers a classically deterministic but turbulent system, for example one satisfying hydrodynamic equations in the turbulent sector, it is no longer possible to measure the velocity v(x,t), since the high frequencies are necessarily unresolved. And even with an arbitrarily higher but fixed resolution, the influence of still higher frequencies is significant. In fact, the field measured by an engineer in a wind tunnel is always an approximation to the expectation value <v(x,t)> of a random variable v(x,t) satisfying the field equations. The values obtained in a single measurement are therefore approximations to the 'renormalised' <v(x,t)> and not the 'bare' unobservable v(x,t).

Since <v(x,t)> is very irregular, and can hardly be predicted, an engineer is interested not in this, but in a to some extent predictable coarse-grained velocity, which he obtains from the average of many instantaneous field values:

v̄(x,t) := <<v(x,t)>>emp = 1/N ∑k <v(x+zk,t+sk)> ,

with small displacements zk, sk. This average velocity can to some extent be well predicted by simulations and is therefore of practical significance. It is also of practical importance to obtain information about the deviations

d(x,t) = <v(x,t)> - v̄(x,t).

This is contained in the correlations

<d(x,t)d(x',t')>emp = 1/N ∑ d(x+zk,t+sk)d(x'+zk,t'+sk)

and higher moments. Thus one sees that there are two stochastic levels: the empirical one, which does normal statistics with the measurements, and an underlying objective one, in which the expectation value can no longer be interpreted as a statistical mean value, but is rather a mathematically defined mass, which makes observable 'renormalised' variables <v(x,t)> out of unmeasurable 'bare' observables v(x,t) containing components of arbitrarily high frequency. This is described in more detail in:

H Grabert, Projection Operator Techniques in Nonequilibrium Statistical Mechanics, Springer Tracts in Modern Physics, 1982.

Another frequently-cited relevant work (which uses diagrammatic techniques borrowed from quantum field theory instead of projection operators) is:

PC Martin, ED Siggia, HA Rose, Statistical Dynamics of Classical Systems, Phys. Rev. A 8, 423-437 (1973).

Interestingly (and unfortunately leading to the traditional confusion) both forms of expectation value have the same mathematical properties, although they express fundamentally different things. Turbluent classical systems no longer have any bare observables (they are just as ill-defined as in quantum field theory) but only renormalised ones. But to describe the dynamics and correlate it to experiment, one needs both. (For more background see turbulence renormalisation.)

This situation is almost completely the same as in quantum mechanics. The only difference is that in quantum mechanics the bare observables cease to commute classically, and become operators in a Hilbert space. In relativistic quantum field theory one finds that the influence of unmeasurably high frequencies is in some sense infinitely large, so that the renormalisation problem leads to considerable difficulties.

I hope to have made clear in this way how the thermal interpretation is to be understood. The objective variables are the renormalised expectation values <f> of the bare field operators f, in full analogy with the classical situation outlined above.

Certain macroscopic operators S(ω) of the measurement apparatus have renormalised values <S(ω)>, which correlate with properties such as the spin of a single particle in experiment ω, and are used as pointer variables. If one performs a measurement N times in random, independent experiments ωk, the distribution given by the Born rule for prepared single particle states is roughly the same as the distribution of the measurable, renormalised <S(ωk)> (and not, as claimed by the Copenhagen interpretation, the distribution of eigenvalues of the particle state allegedly obtained by random state reduction in experiment ωk).

This is the non-trivial point whose demonstration requires the projection operator formalism of statistical mechanics.

Arnold Neumaier (
A theoretical physics FAQ