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I've always been wondered that people speaking of EPR paradox formulate it only in terms of non-relativistic quantum mechanics and not quantum field theory which being a relativist generalisation of quantum mechanics seems to be a more fundmental theory. So, my question is how one formulates the EPR paradox in terms of QFT?

Further reading would also be much appreciated.

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  • $\begingroup$ Prepare the field in a finite time-like region in an "entangled state", let it evolve in time and measure it again in two space-like (w.r.t each other) finite regions. $\endgroup$ – Andrea Becker Aug 21 '16 at 8:41
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EPR paradox concerns pairs of particles. So quantum fields are not necessary. At least you may introduce fields since one particle states are obtained by applying the field operator to the vacuum state. Relativistic quantum mechanics is sufficient since the involved energies are so small that no pair creation takes places. The fact that the theory is relativistic or Galileian is not crucial. What is fundamental is that the pairs of spacetime events where measurements of spin are localized are spacelike separated. The description of the procedure is a just bit different in Galileian and Poincaré case, essentially because the position operator has different definitions and properties in the two cases.

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We do not need to generalize the EPR paradox to relativistic quantum mechanics/field theory, as the only assumption that is needed from relativity is that information travels at a finite speed, due to which we can in principle construct spacelike intervals, which is necessary for formulating the paradox. You can in principle take two non-commuting field operators and do the same thing, but that will not give you anything extra apart from the non-relativistic formulation.

The paradox as formulated in the original paper, was intended to show that quantum mechanics was not a physically complete theory. The intended motivation was that you can measure two different non-commuting observables simultaneously on spacelike separated particles, and since the information can't travel faster than the speed of light, there must be some kind of classical local hidden variables underlying quantum mechanical description which can explain these correlations.

The paradox led to formulation of Bell's inequalities for verifying whether the correlations in a system are classical or quantum. Experimentally classical correlations have been ruled out, and that means nature can't be described by a local hidden variable theory.

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    $\begingroup$ The OP's question was not about WHY we don't generalize the EPR paradox to QFT, but about HOW to describe the EPR paradox in a QFT setting :) $\endgroup$ – Andrea Becker Aug 21 '16 at 9:23

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