The Einstein-Podolsky-Rosen argument has been at the origin of a long debate on the validity of Quantum Mechanics and on its interpretation. It was presented in 1935 and states that Quantum Mechanics is not a complete theory. This argument looses its relevance in a deterministic theory based on the materion, where the measured quantities (position, impulse, energy, spin...) do not correspond with "physical states" but with measurement results which may not logically be determined. In this context, Quantum Mechanics is closer to intuitive ideas than the classical point of view.
Recall of the origin of the EPR controversy.
Quantum mechanics (and therefore its interpretation) was mainly developed in the 1925-1930 period. It did not seem to be a satisfactory theory to describe "Physical Reality": the quantities that are linked to fundamental particles (photons, electrons, protons) or simple aggregates of particles are not always precisely determined. Yet, the 1920's physicist associates "Physical Reality" to the measured quantities.
This feeling of unsatisfaction weighs on the scientific community, without being clearly formulated. That is the reason why Einstein and two of his collaborators, Podolsky et Rosen (further named EPR), set out a reasoning in a paper entitled "Can quantum-mechanical description of physical reality be considered complete?" . In this paper, EPR define at first the conditions that a theory must fulfil to be satisfactory. These conditions are:
EPR do not doubt the validity of the first condition: quantum mechanics is correct. It is in perfect agreement with the experimental reality. Its predictions are right.
According to EPR, quantum mechanics do no satisfy the second condition: it is not complete and therefore not satisfactory.
Argument advanced by EPR
One of the implications of quantum mechanics is: it is impossible to determine accurately a quantity associated to a particle (EPR give the example of the momentum) once one has measured precisely a conjugated quantity (in this example, it is the position).
The argument, according to which quantum mechanics is not a complete theory, is obtained by the following thought experiment. Two particles A and B interact and separate. Quantum mechanics give us the means to predict statistically the results of position and momentum measurements. These predictions are obtained using wave functions. EPR consider that a measurement result corresponds to a part of "physical reality". When the measured position is x1, it is not x2. But as stated above, quantum mechanics imply that it is impossible to measure simultaneously position and momentum of particle A.
In the EPR thought experiment, there exists a diverted way to determine accurately position and momentum of particle A. The momentum of B is measured to obtain the momentum of A. Indeed, this is possible using the law of conservation of momentum of the system A+B. In this way, the momentum of A is known accurately without disturbing its position. Measuring precisely this position makes it possible to know accurately both position and momentum, which was declared impossible by quantum mechanics. EPR therefore conclude that quantum mechanics is not a complete theory.
When presenting this argument, EPR took several precautions like defining "Physical Reality" or the notion of "separated particles". The EPR argument has initiated interesting conceptual and experimental developments like non-locality or separability (cf. other EPR web-sites).
The EPR argument and Materion physics
As seen before, EPR considered that the measurement results correspond to the "real state" of the particle. According to them, a measured position x1 must determine entirely the position of the particle. It is unequivocally x1 and not for example x2.
The materion is a model for an elementary particle: it is a small stick of constant extent without any other intrinsic quality (mass, charge,...). In the conceptual framework of Materion physics, having measured a position x1 does not imply that this position is not equal to x2. Indeed, as the particle has non-zero extent, position x1 only corresponds to the measurement result obtained by the interaction between the materion and the measurement apparatus. This result could also have been x2 if the interaction between the materion and the apparatus had been localized at another point of the materion. There is indetermination in the measurement result of position. In this point of view, quantum mechanics is a theory that is closer to intuition than classical mechanics.
The situation is identical for other measurable physical quantities, like momentum, energy or spin. For momentum and energy, the measured quantities depend on the duration of the interaction between materion and apparatus (i.e. on the localization of the impact point on the materion), which is undetermined. The measurement result of momentum could therefore be q1 as well as q2.
Spin is a more complex physical quantity. One compares it often to an intrinsic angular momentum, i.e. to a vector directed along the rotational axis whose value is proportional to the extent of the particle multiplied by its rotational velocity; the proportionality factor includes the "mass" of the particle. For particles with non-zero charge, this spin generates a magnetic dipole moment which may be detected with the help of a "Stern-Gerlach" apparatus. It is commonly accepted that this spin has a constant direction and therefore there exists a "real spin state" of the particle. This opinion is the consequence of a profitable generalization which compares the rotation of an elementary particle with the rotations that exist in the macroscopic world (gyroscope, planets), i.e. where the direction of the rotational axis remains generally constant (in a good approximation).
This generalization is not necessarily correct in our case. Intuitively, it seems difficult to accept that the rotational axis of an elementary particle, whose destabilizing interactions with the surrounding particles (electrons, photons,...) have much more effect than in the macroscopic world, may keep a constant direction. The measurement result of the particle spin corresponds to a snapshot, which may be obtained through the stabilization of its direction with the help of a magnetic field, like in a Stern-Gerlach apparatus. For the same particle, probabilities to measure a positive or a negative spin are equal. In this point of view, the predictions of quantum mechanics are closer to intuition than the classical point of view of a spin with constant direction.
The EPR argument is based on the preconceived idea that physical quantities reveal unequivocally the "real state" of a particle. Yet, according to the materion model, these quantities (position, momentum, energy, spin) are not intrinsic properties of the particle but undetermined measurement results. The indetermination in the measurement is a function of the sole intrinsic quality of the particle: its extent. In this model, the EPR argument is therefore not correctly founded.
 Einstein A., Podolsky B. and N. Rosen, 1935. "Can the Quantum-Mechanical Description of Physical Reality be Considered Complete?" Phys. Rev. 47 (1935) 777.