roughly 2 events in A and B are simultaneous for an observer in C if the variation of a physical quantity in A and B propagating at given, finite velocity is seen in C like 2 events with distance 0 in 4d Mink. space

Simultaneity is strictly math, not physics. If x^2 + y^2 +z^2 = r^2, then everything that happens on the 4d surface r^2-t^2 = c where c is a constant is simultaneous. This surface projects on XYZ as a sphere and on any 2 space dimensions and time as a hyperboloid of revolution. This surface doesn't have thickness: any point off the surface even infinitesimally isn't simultaneous. But simultaneity can be thought of as having duration and extension in the following sense. If one changes t infinitesimally to t+dt, one can change the location to any point on the sphere R^2 = c+(t+dt)^2 and still be simultaneous.

Within the context of quantum mechanical measurements the notion of simultaneous measurement is a pretty obsolete one. When measurement of incompatible observables is discussed, it is better to refer to joint measurement than to simultaneous measurement when the impossibility of such measurements is at stake, because the temporal aspect is less important than the mutual exclusiveness of measurement arrangements. In particular observation of wave and particle aspects in interference experiments is requiring different measurement arrangements.

@Willem de Muynck

The question is not about what is fashionable. If it is true that all our judgements on time can be reduced to judgements about simultaneity of events, the statement that simultaneity is obsolete would mean that all judgements about time are obsolete, then either time is obsolete or time doesn't exist. However, I still perceive some temporal order.

My remark was not about obsoleteness of time but about the irrelevance of simultaneity in quantum measurements like the EPR measurement (better: EPR-Bell measurement). In such coincidence measurements time does not play a crucial role

because measurement results are independent of whether the two measurements performed in coincidence are carried out simultaneously or not.

@Claus Janew

If time doesn't exist as a fundamental entity, then our ideas derived from the concept of time are not fundamental. In particular the concept of frequency, as a number of oscillations in a unit of time. That won't be fundamental. Apparently our ears can perceive or measure short periods of oscillation, as demonstrated by the fact that we can enjoy, for example, classical music, where consonance appears as a result of a relation between frequencies (Helmholtz natural scale). But we also perceive colour and there is the relation E = \hbar \omega, where frequency is involved and therefore time. Our definition of a second using atomic clocks would be meaningless then?

My question about the simultaneity of events in close proximity is related to the fact that in the classical theory of probability we have the concept of mutually exclusive events, which, it appears to me, can only be considered as mutually exclusive because they cannot simultaneously occur. We are in the same situation in the case of mathematical statistics. Considering that quantum phenomena, like electron diffraction, and the associated randomness are observed when the behaviour of systems in small regions of space is considered, I wonder if the use of complex amplitudes and the principle of superposition of those complex amplitudes appears in some way by necessity, because the concept of simultaneity, as defined by Einstein, is meaningless at an atomic scale and the classical concept of mutually exclusive events is not applicable at that scale.

I would appreciate your remarks on these ideas.

@Claus

I find your answer appealing, because apparently you have an insight on things I do not understand. Would you mind to explain it further or, at least, give some references to consider?

It has been claimed before that time doesn't exist. In previous views it has been explained as an abstraction of all of our perceptions of moving bodies. I think though the same claim may be made for any of the abstractions we use to describe moving bodies: position velocity, momentum, etc. Isn't physics just a way agreeing on how we perceive moving bodies. So the question is, does this have anything to do with physics.

I think it has to do with physics. The theory of relativity emerged from a critical examination of the concept of simultaneity, after the apparently paradoxical discovery of the constancy of the speed of light. But Einstein though experiments referred to physical phenomena in the realm of macroscopic experience, where it makes sense to speak of a system of reference as a body of reference. What body of reference can be considered to provide an operational definition of simultaneity at the microscopic level of atoms and molecules? That might be the clue to understand, at least, why our classical concepts do not work at that level.

Oscar, you justly refer to Einstein's EPR experiment by which he tried to prove the incompleteness of quantum mechanics. His point of departure in this attempt was the so-called `element of physical reality' which should exist independent of any measurement. Einstein thought that an object must have all its elements of physical reality simultaneously. He associated a measurement result with every element of physical reality, and proved that, due to incompatibility of quantum mechanical observables, this is impossible. The important factor here is not simultaneity but the impossibility of jointly measuring incompatible observables. A general proof of this is given much later by Kochen and Specker.

@Willem de Muynck

My reference is to the definition of simultaneity as it appears in A. Einstein; "On the Electrodynamics of Moving Bodies" & "The Meaning of Relativity." In the first reference, Einstein explains "If a point is at rest relatively to this system of co-ordinates, its position can be defined relatively thereto by the employment of rigid standards of measurement and the methods of Euclidean Geometry an can be expressed in Cartesian co-ordinates..." Then he explains that two clocks at rest at different points synchronize if t_B-t_A = t_A'-t_B (The details on the definition of the times t_A, t_B, and t_A' can be seen in the original paper.) All rigid bodies in our experience are made of atoms. In other words, this definition of simultaneity applies in the macroscopic domain of our common experience and it can only be extrapolated to the simultaneity of events in close proximity (at distances of the order of atomic dimensions) by a metaphysical assumption that cannot be grounded on any experiment, because there are no "rigid bodies" the size of an atom, or the size of the separation between atoms in a crystal.

Though I did not refer before to the EPR argument I would like to add that this is an argument by reductio at absurdum and:

“We must be certain that, in showing that the negation of a statement we wish to prove leads to a false conclusion, we use only premises known to be true as well as a logically valid argument. For if the negation of our statement is not the only statement whose truth value is uncertain, or if the argument used is not logically valid, then we could arrive to a false conclusion, even if the negation of our statement happens to be true.” M. C. Gemignani, Basic concepts in mathematics and logic (Addison Wesley 1968, Dover 2004).

Einstein Podolsky and Rosen, in order to arrive to their conclusion, that quantum mechanics was not a complete theory, introduced an assumption: "If without in any way disturbing a system, we can predict with certainty (i. e. with probability equal to unity) the value of a physical quantity, then there is an element of physical reality corresponding to this physical quantity." The only argument they gave in support of this assumption was that "we regard (it) as reasonable."

In strict logic, they should have considered: the axioms of quantum mechanics, a clear definition of a "complete theory"; the assumption that quantum mechanics was a complete; and come to a contradiction as a logical conclusion, without introducing anything else (i. e. this obscure statement about "elements of reality" ) even if they considered it "reasonable." That's how we prove that the square root of two is not a rational number: we start with the axioms of number theory; a clear definition of a square root; and the assumption that there is a rational number equal to the square root of two. Then we come to a contradiction as a logical consequence.

I studied this problem in more depth in my paper "Logical Refutation of the EPR argument." http://physicsessays.org/doi/abs/10.4006/0836-1398-26.1.21 It is my opinion that this EPR argument was not appropriately addressed by Bohr et al, with a devastating effect on further advances in theoretical physics. However, not everybody shares that opinion with me.

Well I think what I am getting at is that it might help to think about experimental tests that could reveal something new. Take your example in your original question: the double slit. If you assume there's a sensitivity to simultaneity here there's only a couple of parameters in the experiment that you could play with to test simultaneity. That would be the distance between the slits and the angle they make with the source. Varying the width would test the distance aspect of simultaneity and varying the angle would test both the distance and time aspects. This is perhaps a hokey example: it's probably been done and probably doesn't show anything. I think the wave nature of the source particles or photons would explain the results. But maybe you see what I'm getting at. If there's nothing testable then it's not physics.

In a way I agree with you. However, there is not point in making experiments when there is not a theoretical framework for the interpretation of the results. How can we put synchronized clocks, and detectors to test if those events are mutually exclusive? Furthermore, if we could do that, we will be able to determine through which slit a particle passes an every one knows, or pretends to know, that this will destroy the interference pattern. My point is that the concept of simultaneity and therefore the concept of time cannot be operationally defined at an atomic scale. The extrapolation, therefore, of space time at that level is a mathematical fiction. We are forced then to conclude with Bohr that there is not a quantum world, at least not a world that can be described using the fundamental concepts of relativity. There are no matter waves. The existence of electromagnetic radiation is confirmed at a macroscopic level. There must be, in consequence an essential difference between matter and radiation, one that goes beyond the simple fact that photons are massless, and has not been captured by quantum theory.

I completely agree with Charles Francis's first sentence. I go a step further than he does in his last sentence: I consider what is described by (the mathematical formalism of) quantum mechanics and relativity theory as referring to (pointer positions of) measuring instruments, not necessarily directly referring to something existing behind these phenomena (i.e. the pointer positions). This does not imply that I deny existence of a reality behind these phenomena. But in my view these are not described by quantum mechanics (nor by relativity theory). Time is `what is indicated by the hands of a clock'. There may exist different, and even deeper, concepts of time (actually those of relativity theory and classical mechanics are different:relativity of simultaneity in RT, not in CM), but they may belong to different theories.

It is well-known that young Einstein reasoned in line with this empiricism,

whereas in his later years he more or less abandoned this view. In 1935 this badly influenced his attempt to prove the incompleteness of quantum mechanics by seducing him to equate results of quantum mechanical measurements (which are in the domain of application of quantum mechanics) with `elements of physical reality' (which are not in that domain, but in the domain of a subquantum theory that is still lacking). As far as quantum mechanical measurement results are emergent, they are so as properties of the measuring instrument, not as properties of the microscopic object.

Well the double slit is a macroscopic experiment: every quantity of interest is measureable macroscopically. With no theory at all the experiment seems to say that massive particles behave the same as those that don't have mass, that is, those that have only wave properties.

Perhaps there is something else the experiment says that makes little demand on theory. It says that when the particles are small enough, some questions can't be answered meaningfully and one of these is "which slit?" Modern versions of this experiment have pursued this question relentlessly to the point of testing when the observer knew which slit the particle went through. The results are still consistent with if the observer knew, there's no interference pattern.

@James

If you connect the terminals of an antenna to the leads of an oscilloscope you can, at least for low enough frequencies observe an oscillation in the local electromagnetic field. I don't think there is an experiment allowing us to observe the oscillations of a "wave function."

@Charles

I know your work and I think I am starting to understand it. However it will take longer. I wish I were younger and faster.

Aren't wave properties shown by simple diffraction: what we call the bending of light around an edge?

Perhaps the reason waves are inferred is by analogy with sound diffraction. If the particles weren't "fuzzy", having some sort of diffuse edges, then they would behave like particles are supposed to: go in straight lines when missing the edge and get cut off when hitting it. The two slit experiment is a diffraction experiment and the pattern is not analogous to single particle scattering off the slit edges. The pattern exhibits reinforcement and extinction. No single particle statistical pattern can explain extinction.

I thought it was agreed that this experiment is macroscopic, that is, classical. All you can do with the single particle patterns is add them and no way of adding then can produce an extinction. Reinforcement and extinction are wave properties. This is just telling the outcome of the experiment, no interpretation is being used. Even if you invoke a special Hilbert space, you still have to exhibit wave properties.

That's why I asked if the concept of simultaneity can be defined for events in close proximity: the electronic distributions (in accord with mathematical statistics) have to be added if the events (passing through a slit or the other) are mutually exclusive; however two events are mutually exclusive if they cannot occur simultaneously. From the point of view of classical mechanics, a particle cannot be in two different positions at the same time (simultaneously).

Well and I gave a mathematical answer, not seeing the double slit as having an issue with simultaneity because the whole experiment is in the "lab" frame, that is, a single frame. Instead even if you had point particles, another quantity is essential: your uncertainty about the exact position of the particle once it's emitted. It's an unstated assumption that by the time the particle gets to the slits, you don't know which slit it goes through. If in fact you had sufficient control of the particle beam to aim at one slit, the there's no interference pattern. So your uncertainty is an essential aspect of the experiment. This is angular uncertainty, as described, but probably you have uncertainty also in the particles velocity, in which case there is also uncertainty about the time the particle gets to the slits. In general the particle should be thought of as having a volume of uncertainty in which it travels.

There exist more general arguments that support the universality of wave nature and have to do with the definition of material world in the basis of stability: That is the common denominator of all physical processes which implies the 2nd order differential equations describing almost all (example: Schroedinger eq.). Besides this, the de Broglie wave length was the starting point of everything in qm, why it took another direction is another topic for discussion.

To return to the original question: We have to define first the term of 'simultaneity' for a single particle-wave, like the electron. Is this possible in principe?

To Charles Francis' comment I would like to add that, although the Copenhagen interpretation is pretty obsolete by now, Bohr's idea of contextuality (in the sense that the measurement arrangement plays an important role) is still crucial in understanding quantum mechanics. Quantum measurements cannot be understood without taking into account the role of the measurement arrangement. The aspect of mutual disturbance in a joint measurement of incompatible observables, which already occupied Heisenberg in the early days of quantum mechanics, is still now an important subject,

Let's put it this way: the entire description of the experiment, including the out come is classical. That's why it's surprising when the pattern can't be explained by classical arguments. Explanations are forced out of the classical mode. But when you see the pattern exhibiting wave properties and having the same kind of pattern for both particles with rest mass and photons, you have clues to the explanation. This experiment by itself doesn't force you to go all the way to quantum mechanics.

There actually is something more to it than is mentioned by Charles Francis. Within one single measurement arrangement measurement results satisfy classical probabilistic laws. Measurement arrangements of standard observables (corresponding to projection-valued measures or hermitian operators) are special cases of more general measurements corresponding to positive operator-valued measures. Among these latter there are joint nonideal measurements of incompatible standard observables the probabilities of these being classical, but yet yielding (nonideal) information on the probabilities of the incompatible standard observables. Only when it is attempted to bring together probabilities of incompatible standard observables in one probability space, classical arguments are no longer applicable. Bohr referred to this as `mutual exclusiveness of measurement arrangements of incompatible (standard) observables'. Contrary to Bohr's dictum measuring instruments cannot be considered to be classical. They need to have a part that is sensitive to microscopic information, which part should be decribed by quantum mechanics. Positive operator-valued measures directly follow from the quantum mechanical description of the interaction of this part of the measuring instrument with a microscopic object.

Well I think the logic goes this way: whatever theory you might use, it can only at best be compatible with an experimental result. The experiment doesn't belong to quantum mechanics, quantum mechanics can only agree with it or not. And there's only the one experiment but many quantum theories which must agree with it or be discarded. The fact that classical explanations can't work only forces you to discard that theory: it doesn't make you choose a quantum theory (and which one?) but what ever theory you propound has to agree with the experiment.

Depends on what you mean by a theory. I think most would agree that what we have presently for quantum theory candidates are by no means complete theories that don't have serious challenges and as far as I know none are derivable from axioms. You mention interpretations which might include rules of application. None, I believe, have unchallenged rules of application. Quantum mechanics itself begins from classical theory and imposes quantization. That is an unsatisfactory beginning: starting from the very thing it seeks to replace. What might be called Copenhagen quantum mechanics doesn't deal with a detector apparatus so what we get ad hoc is an apparatus that's half quantum and half classical. Recall Feynman's famous dictum, something like: anyone who thinks they understand quantum mechanics, doesn't.

At any rate this is all beside the point, which, I think, is that the double slit stands as an experiment completely describable classically but which must be explained by any candidate theory including the observed wave properties. Call the wave properties by any other name, the reinforcement and extinction remain to be explained by whatever theory.

I can respond in part, from the viewpoint of causal set theory, to the questions themselves:

" Is simultaneity a macroscopic concept? Is time and duration, as a consequence, an emergent, and not a fundamental concept?"

With causal set theory, time is definitely a fundamental concept, since it is the ONLY fundamental concept.  Time is characterized as next-to-next succession, so that moments are connected by discrete time steps, or "causal links."  Temporal sequence can fork and re-converge, forming a causal web to serve as space-time.  Such a manifold is composed of primitive time-ordering relations alone-- primitive spatial relations are excluded from the theory.  By excluding primitive spatial relations, simultaneity itself is excluded from physics, for what is a "spatial relation" but a simultaneity relation?  Such is the radical nature of causal set theory, which has no space or geometry, but only the topology of time sequence.  The reduction of space-time to time alone, as the interpretation of Special Relativity, is so radical that Russell and Whitehead could base their solution to the mind-body problem upon it.

http://temporalsuccession.com/

Dear Carey, try to explain your next:

"...For any three moments, if a is earlier than b, and b earlier than c, then a is earlier

than c."

from http://vixra.org/pdf/1006.0070v1.pdf 

to a system of three relatively moving inertial frames.

The transitivity of earlier and later is part of the definition of causal sets.  Motion is not defined until particles have been constructed as causal sets, at which point the relative motion of two or more constructed particles can be defined.  Any assemblage of causal sets may have a dominant proper time axis, and serve as an inertial frame.  In any case, "moving inertial frames" is a concept that is separated by many steps of development from the foundation of causal set theory, where discrete time is defined.  I'm not sure that "three relatively moving inertial frames" is definable from causal sets, or if it is, that it would be recognizable from a traditional point of view, since matter and motion are not fundamental in causal set theory.  Instantaneous spatial relations are excluded from the theory, and thus, geometry is forfeited altogether in favor of the purely sequential topology of time sequence patterns.