Definition of an Ensemble

[A. Neumaier]

On the one hand you say the observable fluctuates iff the state doesn't satisfy $\hat{O} \hat{\rho}=o \hat{\rho}$. On the other hand you say it doesn't. Also I don't see what's so special about the energy and the Hamiltonian as its representing operator.

I was saying that

1. ''fluctuates'' is just a popular buzzword for ''is uncertain''.
2. This has nothing at all to do with fluctuations in space or in time.
3. This can be seen by considering the observable $H$.

1. defines how I am using the term; you had asked for it in post #212. I had also defined the meaning of ''uncertain'' in post #190.

2. follows already since the English language doesn't require the corresponding connotation:
Simple definition of uncertain (from http://www.merriam-webster.com/dictionary/uncertain):

• not exactly known or decided
• not definite or fixed
• not sure
• having some doubt about something
• not definitely known

3. Is a physical instance since it is clear that $H$ is invariant in time and under translations. The fact that $\rho$ is not an eigenstate implies that the value of $H$ is uncertain hence fluctuates, while time and translation invariance imply that the value is constant in space and time. Thus the value of $H$ is uncertain although there are no temporal or spatial changes (and fluctuations in the temporal or spatial sense are absent).

 

[A. Neumaier]

To measure the energy of, e.g., a particle you have various choices. One is to use a calorimeter. The particle gets absorbed and you measure how the temperature of the calorimeter changes.

This is not a measurement of $H$ but of the kinetic energy only.

 

[N88]

This statement is correct with probability 1 - since probabilities can be zero it is a triviality.

Dirac in his Bakerian Lecture (1941) [ http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.205.4474&rep=rep1&type=pdf ] refers to negative probabilities. So do others (including Feynman).

Supporting your interest in eliminating subjective and loose language from QM, how about this:

Tests on ensembles give us an "average value". Let our subjective anticipation of that average value be the "expectation".

Associated with the objective "average value" we can derive the objective "prevalence" of relevant observables, which is never negative.

Associated with the "expectation" is the subjective (and often confusing) notion of "probability". That it can be anywhere negative is up to the subject subject and beyond me. Objectively, use of the term "prevalence" is a step toward eliminating all debate re "probabilities" (and "betting") in physics.

 

[A. Neumaier]

negative probabilities.

Statements about negative probabilities are correct with probability $<0$. I.e., there is not even a set of measure zero where these statements are true (excpept perhaps in a figurative sense).

 

[A. Neumaier]

"prevalence"

Essentially nobody understands the meaning of this word. I find it unacceptable to try to solve philosophical issues by introducing new words without a clearly defined meaning.

 

[N88]

Essentially nobody understands the meaning of this word. I find it unacceptable to try to solve philosophical issues by introducing new words without a clearly defined meaning.

Referring to post #228, and seeking clear definitions: A probability assignment (symbol = lower-case p, say) is a normalised subjective judgment based on incomplete knowledge about an ensemble of interest. A prevalence (symbol = capital P, say) is a normalised objective fact about a fully tested ensemble.

Let a fully tested ensemble have sample space Ω with numerical observables O_i. Then the mean value of the observables is:

⟨O⟩ = 1/N ΣN_iO_i = ΣP_iO_i​

where P_i is the prevalence of observable O_i, the normalised proportion of O_i in the tested ensemble.

Thus, prevalences P and probabilities (substituting p) satisfy the same rules:

1. P(A|A) = 1 = P(Ω).
2. 0 ≤ P(Α|Β) ≤ 1.
3. P(¬A|B) = 1 - P(A|B).
4. P(AB|C) = P(A|C)P(B|AC).​

But each P represents an objective fact, each p represents a subject judgment.

 

[A. Neumaier]

But each P represents an objective fact, each p represents a subject judgment.

So you just invent a strange new name ''prevalence'' for the standard notion ''relative frequency''. But what does it mean that an incompletely tested ensemble has a prevalence of $\sqrt{1/2}$, say? Note that most ensembles in physics are infinite, hence can never be exhaustively tested in your sense.