Position and Momentum are pontryagin duals which means there is a basis transformation involved between them with a fourier transform relationship between the basis. As established many years ago (Benedicks’s theorem, Hardy’s uncertainty principle, etc), a nonzero function and its fourier transform cannot both be sharply localized. It therefore makes no sense to talk of position and momentum as being simultaneous elements of reality for the same particle.
In fact, it is well known that it is impossible to make a device that can measure both position and momentum at the same time. In other words, momentum is not well defined at a given position for a given particle, and vice versa.
Lord Jestocost
Read my comment “Lord Jestocost December 5, 2016 at 5:07 pm”
A physical quantity is a physical property/feature of a physical system that can be quantified by measurements, you get a value.
The physical system can and does include the measurement apparatus.
Lord Jestocost
A “element of reality” is a physical property which can be quantified – following Einstein’s operational measurement procedure – “without in any way disturbing the system”.
No! According to EPR, if you can predict the value of a physical quantity “without in any way disturbing the system”, then that *physical quantity* CORRESPONDS to an *element of reality*. Nowhere do they say the *physical quantity* itself is an *element of reality*.
Lord Jestocost
Physical events or measurements have nothing to do with the notation “element of reality”.
Measurements just reveal the values of “elements of reality”.
This is false. Many physical events and measurements *create* elements of reality. Most often, the physical quality is a result of an interaction between the measurement equipment and the “system”. EPR only say the *prediction* has to be done without disturbing the system, NOT that the physical quantity be obtained without disturbing the system.
Lord Jestocost
#1) In case you have proved once that the velocity of a moving electron is an “element of reality”, you have proved it for all moving electrons. An electron is an electron! And velocity remains an “element of reality” for all moving electrons, no matter what you are planning to do or what you are really doing with the electron.
According to EPR, the velocity of a moving electron CORRESPONDS to an “element of reality”. But not all elements of reality are independent of the measurement process. For some measured physical quantities, the element of reality to which it CORRESPONDS is in fact created as a result of the interaction with the measurement apparatus. It will be terribly wrong then to argue that unmeasured electrons possess such elements of reality. At best, they can possess the potential to produce these elements of reality on measurement. Your statement above is therefore generally false.
Lord Jestocost
#2) In case you have proved once that the position of a moving electron is an “element of reality”, you have proved it for all moving electrons. And position remains an “element of reality” for all moving electrons, no matter what you are planning to do or what you are really doing with the electron.
See previous point for a refutation of this.
Lord Jestocost
#3) As an electron is an electron, you can conclude from #1 and #2, that both velocity and position are simultaneous “elements of reality” for all moving electrons. That’s the classical definition. Both are “elements of reality” and measurements or physical events play no role.
A Measurement on one “element of reality” does not “retroact” or does not affect the past in this sense that the “before measurement feature” of being an “element of reality” is erased for the other “element of reality”.
Therefore your conclusion is completely false. If the act of measure creates an element of reality due to interaction with a measurement device, while at the same time destroying the original “system”, rendering the second mutually exclusive measurement impossible. It will be impossible for both “elements of reality” which were *possibilities* before measurement, to become actualized. It is therefore false to conclude that both can ever be simultaneous *actualities*. Contrary to your claim, measurement on one element of reality does not retroactively falsify the previous *possibility* that wasn’t measured, it simply renders it *counterfactual* and impossible to actualize. No element of reality is erased in the process. Before measurement, you had two *possibilities*. After measurement, it is still true that you had two *possibilities* before you measured. But in addition, you now have one *actuality* corresponding to one of the possibilities, and another counterfactual possibility.
Note the fact that the measurement can’t be done anymore does not amount to erasure of the possibility. For example, the *possibility*
If I measure at (a) I will obtain A.
expresses the truth of relationship between “I measure at (a)”, and “I obtain A”. It absolutely does not express any truth about “I measure at (a)” alone, or “I obtain A” alone. Measuring at (b) does nothing to negate that truth, nor does obtaining B negate that truth unless you also measured at (a). The part that is rendered impossible after measurement is the antecedent (“I measure at (a)”). Thus even though it is no longer possible to measure at (a), it is still true that “had I measured at (a), I would have obtained A”. This is a counterfactual statement.