Why am I asking this question? Well, the logical principle of explosion states that if a contradiction is introduced in a logical system then any theorem can be proved true. (Ex contradictione sequitur quodlibet.)
In Einstein's "Meaning of relativity" pp. 43 and 44, Einstein wrote "...Let's imagine a body upon which the electromagnetic field acts for a time....We shall assume that the principles of conservation of energy and momentum are valid for the body. The change in momentum \Delta I_x, \Delta I_y, \Delta I_z, and the change in energy \Delta E, are then given by the expressions [some integrals] ...This four-vector can also be expressed in terms of the mass m, and the velocity of the body, considered as a material particle....which refers to an infinitely short portion of the four-dimensional line which represents the motion of the material particle....We see that this four-vector [the four-velocity] whose components in the ordinary notation are... is the only 4-vector that can be formed from the velocity components of the material particle... By equating the components we obtain in three dimensional notation: I_X = m q_x/\sqrt{1-q^2},..., E = m /\sqrt{1-q^2}. (in other words E^2/c^2 - p^2 = m^2 c^2)"
What is this four-dimensional line representing the motion of this body as a material particle? Is it the center of energy? It cannot be, because the path of the center of energy is not covariant. Actually, years later a non-interaction theorem was proved for Special relativity! ( http://www.physicsforums.com/showthread.php?t=225026 ) The definition of a "center of mass" within the limits of special relativity, representing the motion of a system of particles as a whole, in the way of classical mechanics, is an open problem today. ( http://arxiv.org/pdf/0901.3349.pdf )
The equation E^2/c^2 - p^2 = m^2 c^2 was used by de Broglie in his Doctoral thesis; the belief that it is true is behind Klein-Gordon's equation and the commutation rules of the gamma matrices that appear in Dirac equation; and Dirac's electron theory is foundational for QED! (Michelle Maggiore; A Modern Introduction to Quantum Field Theory; (2005).)
If the equation E^2/c^2 - p^2 = m^2 c^2 is false then, I am sorry, but our beautiful quantum mechanics is just a mathematical game.
"considered as a material particle." It is not an extended body but a massive material point. The electron and other fundamental particles are point particles, even in quantum mechanics. For an extended (rigid) body, there are other forces than the electromagnetic one, i.e. internal ones between the particles composing it.
Dear Oscar Chavoya-Aceves,
a point-like particle is an idealization, and Einstein uses this picture to explain some basic insights into the theory of special relativity.
However, the energy-momentum equation you are referring to is a much deeper consequence of the Poincare symmetry of natural laws in flat space time, i.e. of the semidirect product of the space-time translation group T_4 and the proper orthochronous Lorentz group SO^+(1,3) or its double cover SL(2,C).
In quantum field theory, physical states become elements of an abstract Hilbert space or "wave functions". These objects obey transformation laws which correspond to your equation E^2/c^2 - p^2 = m^2 c^2, in a way every "Fourier component" which builds up a complicated object like a physical electron (consisting also of non-localized (virtual) particles like quarks and an accompanying Coulomb field etc. etc.) has to follow a relation comparable to the naive energy-momentum formula.
You have to couple relativity to quantum theory = quantum field theory in order to discuss problems of the type above.
@Andreas Aste
Can you please explain how this argument avoids begging the question, considering that the correspondence principle plays a role in the formulation and justification of quantum theory in general and quantum field theory in particular? The Poincare transformations are defined precisely, using that point-like idealization that you mention, by the condition that they leave Minkowsky's space-time interval invariant, from which follows the condition \mu_i \mu^i = 1 for the four-velocity of an observer and Einstein's relation under the assumption that p^i = m c \mu^i. Isn't it the case that we will consider false any quantum field theory predicting something different from E^2/c^2 - p^2 = m^2 c^2?
Dear Oscar Chavoya-Aceves,
it is hard to answer your question becausewe do not have a clearly defined basis for the discussion. Firstly, the non-interaction theorem is based on a picture with a conserved number of individual classical particles, a picture which does not exist in full quantum field theory. Secondly, the Poincare symmetry is a fundament of QFT which has proven to be very fruitful, and it is clear that most theories are based on the assumption that Poincare symmetry holds, at least in flat spacetime. Giving up Poincare symmetry means to give up the classical concept of spacetime, such issues are discussed in theories like quantum gravity, noncommutative geometry etc etc.
Deviations from Poincare symmetry would have severe consequences for particle physics, not only a conflict with "E^2/c^2 - p^2 = m^2 c^2", but signals like e.g. the violation of CPT symmetry have not been observed up to now. At Planck length, we definitely should expect a breakdown of the Poincare symmetry. There is a plethora of hundreds of papers which discuss deviations from "E^2/c^2 - p^2 = m^2 c^2" in the framework of quantized spacetime.
Thirdly. if you insist on Poincare symmetry, the full picture in QFT concerning an energy-momentum-mass relation is less naive than insinuated by your question. An electrically charged particle is an infraparticle, an therefore not an eigenstate of the mass squared operator - an electron is soft, if you accelerate it, it will not behave in the naive way like "a=F/m", be it relativistic or not. Such issues are discussed in papers of Schroer, Strocchi, Fröhlich, Haag, Borchers, Buchholz... anyway, Lorentz symmetry is spontaneously broken in charged sectors of QED, see e.g. http://link.springer.com/article/10.1007%2FBF01206185 . But beware, a broken symmetry does not mean that the underlying natural laws are broken, but the solution of the laws: A particle or a chair are not Lorentz invariant, if you boost them, you get something different, but the laws governing their behaviour remain the same.
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