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The time is treated differently in special relativity and quantum mechanics. What is the exact difference and why relativistic quantum mechanics (Dirac equation etc.) works?

Andreas K.
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    Er...time is treated differently in relativistic mechanics and non-relativistic quantum mechanics, but that is the same as saying that time is treated differently in relativistic and non-relativistic classical mechanics. – dmckee --- ex-moderator kitten Jul 15 '11 at 02:04
  • Quantum mechanics doesn't per se imply relativity. – Siyuan Ren Jul 15 '11 at 03:18
  • The Schrödinger equation of non-relativistic QM is second order in the time derivative and is not Lorentz invariant. On the other hand, the Dirac equation is first order in the time derivative and is invariant under Lorentz transformations. So I think this is the main difference between non-relativistic and relativistic QM. In the latter, the time is treated in (almost) the same way as spatial coordinates. Also the spin is a relativistic effect because it emerges naturally only in relativistic QM. – Andreas K. Jul 15 '11 at 14:33
  • Dear @ANKU: Your above comment the Schrodinger equation of non-relativistic QM is second order in the time derivative was probably written in a bit of a hurry. :-) More importantly, is it possible to formulate the main question using precise terms? – Qmechanic Jul 17 '11 at 15:51
  • Oops, it's second order in the space and first order in time. But the point is this, we make it first order in space and time derivative so that it becomes Lorentz invariant. Right? – Andreas K. Jul 18 '11 at 02:12
  • Note, one may ask instead "The space is treated differently in special relativity and quantum mechanics." It might be better to think that space became more time-like and not time more space-like. Though these are quite vague expressions. – Yrogirg Aug 01 '12 at 13:55

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Quantum mechanics can be reconciled with special relativity to make quantum field theory, but there are some awkward things going on in that marriage. SR treats time symmetrically with position, but in quantum mechanics, position is an operator and time isn't. Baez at UCR has a nice discussion of that here: http://math.ucr.edu/home/baez/uncertainty.html

  • Well, QFT reconciles this by disposing of the position as an operator. – Marek Jul 19 '11 at 20:18
  • And Dirac's approach (and Feynman's) makes time an "operator" (or equivalently an integration variable in the path integral). This answer is no more satisfying than saying "In classical mechanics, position is a function, and time is a parameter". That's true, but only if you choose to parametrize by time and not proper time. The same is true in quantum mechanics. – Ron Maimon Aug 13 '11 at 20:09
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Time is always time. It is special. Another thing is its involvement in transformations of measured data from one reference system to another. This involvement does not change its meaning. In a given reference frame the time is unique and the space coordinates are multiple - according to the number of particles to be observed.

Concerning Dirac equation, it took some efforts to make it work after its invention. It works because it was made work, if you like. Besides, it depends what exactly do you mean by "relativistic QM". QED, for example, is rather difficult to make work. Its sensible results only appear at page 500 or so, when the infrared catastrophe is resolved.

Vladimir Kalitvianski
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