Are Quantum Mechanics and Relativity Logically Compatible Theories?

[PhysicsBuff99]

I've often read that quantum mechanics and relativity theory are logically incompatible theories, meaning that both theories cannot be true. Can anyone use mathematics to prove that?

Thanks,

King

 

[NeutronStar]

Special and General Relativity

You need to realize that here are two theories of relativity, Special Relativity (SR) and General Relativity (GR).

As far as I know there are no inconsistencies between Quantum Mechanics (QM) and SR. The inconsistencies only exist between QM and GR.

Those inconsistencies are in part due to the fact that GR requires a smooth continuum, and QM requires a discontinuous quantum jumping.

Maybe someone else will post the precise mathematical equations that are incompatible.

 

[Hurkyl]

Quantum mechanics is also done on a smooth continuum.

The problem is that while QM is done on a specified geometry (such as proven QM theories, and String theory), it is significantly more difficult for the geometry itself to be part of the theory.

 

[mathman]

When it comes to describing what happens inside black holes, GR and quantum theory, when used together, give mathematically nonsensical results. I believe recent books by Brian Greene discuss this better.

 

[NeutronStar]

Quantum mechanics is also done on a smooth continuum.

Hmmmm? I wonder if the all the people who are trying to explain the collapse of the wave function are aware of that?

Who was it that said? "If we are going to stick to this damned quantum-jumping, then I regret that I ever had anything to do with quantum theory."

I believe that was Erwin Schrödinger

While the Schrödinger equation might be based on a continuous mathematics, the instantaneous collapse of the wave function is not. In other words, while the probability waves are continuous the fact that only one point in the probability wave can actually be considered to be the "actual" answer is where the discontinuity comes into play.

So saying that QM (as a theory) is done on a smooth continuum is arguable at best I think.

Oh, wait! Before I leave,…

 

[seratend]

[

Hmmmm? I wonder if the all the people who are trying to explain the collapse of the wave function are aware of that?

Who was it that said? "If we are going to stick to this damned quantum-jumping, then I regret that I ever had anything to do with quantum theory."

I believe that was Erwin Schrödinger

While the Schrödinger equation might be based on a continuous mathematics, the instantaneous collapse of the wave function is not. In other words, while the probability waves are continuous the fact that only one point in the probability wave can actually be considered to be the "actual" answer is where the discontinuity comes into play.

So saying that QM (as a theory) is done on a smooth continuum is arguable at best I think.

Oh, wait! Before I leave,…

QM uses the euclidian geometry (classical) or the minkowski geometry (relativistic) where the symmetries allow to define the generators (p,j, H, etc …) of the theory. This is the "continuous" aspect of the theory (isotropy and homogeneity and galilean/Loarentz invariance).

The projection postulate does not refute this "continuous" aspect. Considering this, it is quite analogue having a probability space defined on the |R set (continuous) and saying that the conditional probability on a subset of |R refutes the fact that |R is a continuous set (I hope this is the right term).

Seratend.

 

[dextercioby]

I've often read that quantum mechanics and relativity theory are logically incompatible theories, meaning that both theories cannot be true. Can anyone use mathematics to prove that?

Thanks,

King

Treating GR as an ordinary field theory encountered in the SM is practically useless,because the quantum theory obtained is not renormalizable.
Hilbert-Einstein action describes at quantum level a theory of SELFINTERRACTING gravitons,quanta of gravitational field i.e.particles with spin two.
Other attempts have been made of finding something else instead of the HE action.The linearized theory of gravity (developed by Einstein in 1916) is basically "good" when it comes to quantum behavior (i analyzed this theory using standard BRST antiparanthesis-antifield forrmalism (developed frankly by Batalin&Vilkovisky,but it's usually called Lagrangian BRST (cf.the Hamiltonian approach found independently by Becchi,Rouet,Storra and Tyiutin))),but it has the disadvantage of working with the gauge-fixed Pauli-Fierz action which describes the FREE (i.e.NONSELFINTERRACTING) GRAVITONS.In fact,these gravitons interract with ghost fields.Other attempt was the so-called Weyl gravity (i.e. gravity based not on the Riemann curvature tensor (of the curved manifold called spacetime),but on the conformal Weyl tensor,un ugly (still 4-th order) tensor).This theory is excellent at quantum level (i.e.renormalizable) but the classical (nonrelativistic) limit of the unquantized action gives you 4-th order LODE of motion (cf.2-nd order LODE of motion in the Newtonian limit of Einstein GR).
A step forward was made by Elie Cartan who developed the so called "Einstein-Cartan GR" which used other fields (called vierbeins,usually seen as vielbeins) for describing the gravitational field.This theory is good because it allows coupling with spinor and scalar matter fields in a theory called SUPERGRAVITY.If I'm not mistaking,these theories of Sugra (apud Supergravity),though allow an unifying theory of all 4 fundamental interractions,are,at quantum evel,still nonrenormalizable.I mean,if they were (it doesn't matter how many supemultiplets of particles it envolved),why would ST and LQG be alive today??
And then,ST,the final (??) frontiere.It is said to give a satisfactory behavior of gravitational interraction at the quantum level (this time there are no point-particles like in SM anf SUGRA,but strings,10-dimensional objects).

This a plainy simple and incomplete (probably incorrectas well,at least in its final lines) review on the the work that's been done by theorists worldwide in the last 70 years or so in the field of Quantum Gravidynamics (the name i give for the theory of QG in agreement with common names used in the SM).

Daniel.

 

[dextercioby]

And to formulate an answer to the initial question,i think that we cannot conceive theoretical physics today without the two theories in question.And YES,THEY'RE BOTH CORRECT.WE JUST (STILL) LACK THE THEORY WHICH INCLUDES THEM BOTH AS PARTICULAR,LOW-ENERGY,4 DIMENSIONAL THEORIES.