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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?

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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 Jul 15 '11 at 2:04
Quantum mechanics doesn't per se imply relativity. – Siyuan Ren Jul 15 '11 at 3:18
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. – Andyk 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? – Andyk Jul 18 '11 at 2:12

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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:

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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

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.

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Combining General Relativity Theory with Quantum Theory ‘The lifetime of a mass or an energy in space is its Mc2 energy’ Ref.(3). Due to this characteristic feature of a substance, conversions of photons of the wawe-particle (energy-mass) or electrons continue consistently. Hence, the binary conversion behavior of a photon implies binary conversion behavior of a big mass space object. In other words, the behavior of a photon is a miniature version of the behavior of a big mass space object. Because of General Relativity Theory, ‘One hour in the Sun remains behind with respect to, one hour in the Earth. One hour in the Earth remains behind with respect to, one hour in the Moon. One hour in the Moon remains behind with respect to, one hour in Alpha Ray. One hour in the Alpha Ray remains behind with respect to, one hour in Beta Ray. One hour in the Beta Ray remains behind with respect to, one hour in Gamma Ray. They all show the same physical behaviour. Let us observe the behaviors of two photons, say one like is a big mass, and the other like is a small mass. Since the photon with small mass has a short lifetime, it will transform faster from mass into energy and vice versa. The big mass photon has a longer lifetime. Hence, the speed of transformation from mass to energy or from energy to mass is slower. Smaller the mass of a photon is, the much bigger the kinetic energy is. The kinetic energy of a photon is given by, e=hf ‘In order to calculate the lifetime of a mass or an energy in space, we can assume time flow to be time/energy; in any case, no matter what value we assign to time flow, that will not change the present result: the lifetime of a mass or an energy in space is its Mc2 energy. When this is calculated, the lifetime of 1 kg mass in space is 2,851,927,903.26… years, or 9.10 16 s’ Ref.(3) All photons’ and all free sub-atomic particles’ lifetimes are their periods or 1/f . In other words the periods are lifetimes for photons and for free sub-atomic particles. And a period is equal its Mc2 particle energy x 1 s/joule or erg. If the period is high, lifetime is high. And Mc2 is high. Or vice versa. Like astronomical objects. This is universal law. The mass of the low frequency of a photon has a big value . For example: Substituting an Alpha Ray with 1,67.109 Hz frequency into the formula yields, e= hf = (6,62 .10-34) x (1,67.109) time = (energy) x (time flow) t = 1/1,67.109 = Mc2 .1 s/joule ; M = 6,64 .10-27 kg On the other hand, for a high frequency photon the mass has a small value. As an example to show this is the case, consider a Beta Ray with 1,22.1013 Hz frequency into the formula gives, e= hf = (6,62 .10-34) x (1,22 .1013) t = 1/1,22.1013 = Mc2 .1 s/joule ; M= 9,109.10-31 kg These two examples shows us that by equating the period of a photon to Mc2 energy in a unit time flow provides us the actual values of the mass. These two examples are valid for x-Ray ,Gamma-Ray and Light-Ray. When mass decreases, frequency increases. And the transformation from mass to energy becomes more uncertain. We understand that Quantum Mechanics is not different from Classical Mechanics and Relativistic Mechanics, in fact. The characteristics of the particles in Quantum Theory are the same as the character of the mass in the General Relativity Theory. They are subject to the same physical processes. So, Einstein’s expression “God does not throw dice!” is still valid. References (R1): [.1.] Salih Kircalar, ’Utilization of Time:Time Flow’, Galilean Electrodynamics 13, SI 1, 2 (2002). [.2.] Salih Kircalar, ‘Time Effects Caused by Mass or Energy’, Galilean Electrodynamics 15, SI 1,8 (2004). [.3.] Salih Kircalar, ‘Mass or Energy & Quantum Mechanics’ , Galilean electrodynamics 18, J/F, 2 (2007). Salih Kircalar Güzel Otomotiv Kizilelma Cad.No:99/B Findikzade-Istanbul /TURKEY

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This answer (v1) seems to be just rambling text with no relation to the question (v1). – Qmechanic Sep 23 '12 at 19:51

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