Quantum and Relativity

[gc]

Why are Quantum & Relativity theories incompatible? I don't see why they can't both be true, and I would like to have a discussion. Can anyone provide a (relatively) simple example where the two are at odds?

Thanks.

[Severian]

Quantum mechanics replaces Newtonian mechanics when things get really small. Relativistic mechanics replaces Newtonian mechanics when things get really fast. If you want to do both at once (small things going going really fast) you need a relativistic quantum mechanics.

 

There is nothing wrong with this. In fact, I teach a course called Relativistic quantum mechanics to grad students. Everything works fine.

 

The problem is when you try too describe gravity. General Relativity describes gravity at large distance scales but doesn't work (it just makes wrong predictions) at small distance scales. Attempts to make a quantum version have so far failed because we find infinities crop up all over the place and aren't under control.

 

I would regard this as a technical issue though. I don't think that they are incompatible per se. It is just that we haven't got it working yet.

[gc]

Thanks for your response.

I was always told that the two were incompatible (ie that if one is correct, the other must not be entirely correct), but I guess that's not true? The only problem is that we can not explain gravity with either theory? Other than that, the two theories are not incompatible?

[Severian]

Thanks for your response.

I was always told that the two were incompatible (ie that if one is correct, the other must not be entirely correct), but I guess that's not true? The only problem is that we can not explain gravity with either theory? Other than that, the two theories are not incompatible?

 

It depends on what you mean by 'relativity'. Special relativity (the can't go faster than light stuff) and QM are most definitely not incompatible. General relativity (which is theory of gravity) is not compatible with QM.

[fredrik]

In addition to what Severian already wrote, I think one can view these problems from several perspectives.

 

Technical, mathematical or philosophical and logical.

 

General relativity and QM aren't directly in competition, they rather on their own describe different restricted domains of reality. But since there is only one reality, and we do not like the idea of decomposing reality into different domains with un unclear connection, thus we expect that ultimately we should find a unified theory that describes all of reality.

 

The technical point is that if we simply take these two theories, and their set of equations, and mathematically try to plug in and solve them assuming both hold (ie solving QM during the constraints implied by General relativity) it gives nonsensial results and infinites and is thus mathematically ill defined. Indicating that something is wrong and we might need to find another way to calculate it.

 

The philosophical aspect acknowledges that QM and GR are from start formulated on different platforms. For example Energy in QM, is not define the same way as energy in GR. Here there are fundamental problems that need to be solved before the equations from te different theories can even be compared properly. Thus in this view, it really isn't unexpected that the calculations made (if ignoring the fundamental problems) leads to problems.

 

Of course, quantum mechanics has philosophical problems of it's own. But these are often ignored from the technical point of view, becuase it's mathematically well defined and the computation schemes have been perfectly successful, thus for some motivating the ignorance of the fundamental issues.

 

Another aspect is to see that

 

1) QM suggest that state of matters is uncertain due the nature of physical information exhanges.

 

2) General relativity suggests that our reference frame (spacetime geometry) itself is not fixed, it's dynamical.

 

We should then notice that a state of matters, always has to be evaluate relative to a reference. General relativity suggest that this reference itself is dynamical. This leaves us with something very dynamical with uncertain structure. It seems we have not yet wrapped our heads around howto solve this, and describe it with a consistent well behaved math.

 

So not only are reality uncertain, also our reference is uncertain. This is why extreme care must be taken, because this is a bit unreal to imagine by normal intuition. We basically have no universal reference anywhere. Where do we start?

 

/Fredrik

[Severian]

I would also like to add a technical point. Even though theories of quantum gravity, such as supergravity are non-renormalizable, that doesn't make them wrong. It just makes them incomplete. After all, Fermi's theory was also non-renormalizable, and look what that eventually led to...

[Norman Albers]

I am finding much to do and learn around the analysis of the Reissner-Nordstrom metric. Please check me on the units and quantities here: I calculate the Schwarzschild radius of the electron to be 4E-60 meters. On the other hand, the second term in the metric expression, positive in the inverse square of radius, has a characteristic length of 1.3E-22 m. The electric energy overwhelms the gravitational term in the nearfield, so that there is manifest no event horizon. There is a small dip in the graph and it seems strange that I calculate the point at which the [math]g_{oo}[/math] plot again crosses the x-axis is only about a tenth of a millimeter. Now we are not talking about much gravity here, but the scale is most interesting.

[GutZ]

I hardly ever understand what Norman Alber says lol.

[Norman Albers]

The result of all this is that in the far field the gravitational effects of an electron's electric field die off faster than the term supposedly given by its mass. Conversely, as you go in close, the term created by the modelling of electric field energy dominates the gravitation expression, and actually gravitation changes sign! I have not dealt with this previously so it is a wonderland to me. Now most of us agree that our physics falls apart, or comes together, at the Planck length, so we never get to the smaller radii at which electron gravitation becomes major in the small. Remember the "rubber membrane" of GR strong fields? I'm speaking of where the sheet changes from tension to compression for a charged particle.

[Martin]

... Please check me on the units and quantities here: I calculate the Schwarzschild radius of the electron to be 4E-60 meters. ...

 

If not too much trouble, what formula are you using for the Schwarzschild radius?

[Norman Albers]

Martin, thanks, my book has the "geometric mass" or Scwarzschild radius as [math] m=\kappa M/c^2[/math]. The metric result is [math]g_{oo}= 1-2m/r + \kappa e^2/[r^2(4\pi \epsilon_o^2c^4m_e)][/math]. For electrons or any other particle, actually, the Schw. radius is way below the Planck length; the graph in the nearfield is dominated by the electric term and shows only a small dip where the first order term shows. We know the electric force is "much stronger" but this is examining the subtleties. . .I just edited the incorrect power of 'c',and [math]M=m_e[/math]. Note that I am elucidating the original solution. In my thread on 'Reissner...' I develope nearfield possibilities suggested by a spreading out of the region of divergence, or charge density. which change this picture, though not until dimensions at and below classical radius. . . . . . . .more time passes . . . . .ARGHHH, I blew some of the numbers, though the substance of the discussion offered remains. Now I calculate the Schwarzschild electron radius as 7E-58; I had something upside-down and if you figure the radius where the two r terms cancel. Actually this is the point where the graph, which would have been the Scwarzschild (1-2m/r) and falling down through zero from a farfield of 1, goes back up across the r-axis. Twice this radius is where the curve has its minimum. . . . . . .After much numerical chaos I again got the result of a tenth of a millimeter. I got confused trying to simplify the expression to: [math] g_{oo}= 1 - R_m/r + (R_e/r)^2[/math]. The quantity [math]R_e[/math] is about 0.4E-30.

[Norman Albers]

In low-order regimes we interpret this metric as equivalent to the classical gravitational potential [math]\phi[/math] expressed in: [math]g_{oo}=1+2\phi/c^2[/math]. Thus the original Schwarzschild soln. comes down from a farfield of unity, crosses the axis (becomes zero) at r=2m, and continues to tank downwards. This is not the case with the electric energy part of the gravitational field. This should not be confused with fields themselves. To wit, the forces of an electron on another, to this reckoning, are always forty-two magnitudes in ratio. Thus in a sense we are talking of a 'gravitational flea on an electric elephants back'. In the large, electric forces do not seem to be important as they seem to have opportunity to neutralize at small scale. It is, however, correct to say here that gravity has become repulsive, seen as a rising potential going inward, rather than falling. Protons, too, will show this, scaled by their mass being about 2,000 electron masses; neutrons will not.

[Norman Albers]

I gather the Reissner-Nordstron approach to the electron was discarded because it produces this embarassing positive asymptote in the gravitation potential. What I can offer here, if there is any usefulness in the idea of inhomogeneous charge nearfields, is that inside of the millimeter radius there would be a slow rising of the graph of [math]g_{oo} [/math]. Realize that this term is scaled to the characteristic length of E-30 meters as noted above, and is near Planck length. Much inside of the classical radius (E-15)m, I have shown that the "shading function" approach of my electric field, designed to cancel out terms of order <-2,-1> in radius as you go to the origin. also make the electric term in the gravitation metric expression drop out, leaving terms in -1/r. I see here a possible next move in this chess game. The untenable electric infinities have been removed. What has been solved was a gravitational equation for free space, in terms of a "point mass" possibility, but with the additional electric field energy specifically accounted for. Realize that the -2m/r term shows up as a constant of integration, simply. I suspect that if I developed the relativistic forms (Einstein equations) with the total energy densities included, as have done in my studies, then also the -1/r term would be balanced off, cancelled in the limit.

[Norman Albers]

With the next day's perspective, it is not important to see the -2m/r singularity alleviated, because it would exist at a non-physical size. What I see is that just as I was able to plug my electron field into the Reissner-Nordtsrom machinery, I have three other terms to attempt the same: in an inhomogeneous stew one accounts for source terms in charge and current and also for field energy terms. (I do the same when I calculate total angular momentum.) This is mathematically exciting because, say, [math]\rho U[/math] in this model has stronger dependence at the origin than does the square of electric field. I am just mastering the tensor curl formalisms I need to present the magnetic terms. I do not mean to derail this thread. I mean to stimulate discussion of how both quantum mechanics and relativity can and must be moved further, to further synthesis. You might say, who needs a static model of electrons? This is static only if your thinking stops here. In the case of my similar meditations on photons, a PhD candidate has become quite excited about constructing photon position operators, and feels he has the overall structure. He is taking off with his knowledge of QFT, as impressed with my field manipulations as I am by what he can do. I describe what you might expect from a plasma physicist looking at the vacuum as a superconducting availability. He sees the key to constructing a nonlocal representation.

[Norman Albers]

After looking at the construction of the stress-energy tensor for electromagnetic energy, it seems it is valid in the usual expression: [math]T_{ab}=[F_{ac}{F^c}_b + \frac{1}{4} g_{ab}F_{cd}F^{cd}] [/math], even for inhomogeneous fields, namely those including charge and current. I am assuming an entirely electromagnetic identity to the electron so there is no need to identify a mass density term, as if there were particle masses, and furthermore there is no need to run the source term [math]\rho U[/math] through the gravitation equations. The electric field already shows, or implies the charge density field, so I just talked myself out of a job. Not so for the magnetic field, however.

[Norman Albers]

When we witness macroscopic superconductivity, is this describable by the term quantum nonlocality? Severian, it would be helpful if you would elaborate on your comments on normalizability and Dirac....................Oops, sorry for confusing things. You got it below.

[Severian]

Severian, it would be helpful if you would elaborate on your comments on normalizability and Dirac.

 

Do you mean renormalizability and Fermi?

 

Long before the W and Z bosons were discoverd we knew about weak interactions. They cause things like beta decay and influenced the interactions of particles we saw in our colliders at the time (various mesons mainly).

 

We had a quantum field theory for electrodynamics, QED, and we wanted a similar idea for the weak interaction. One formulates QED in terms of a Lagrangian where the fields (particles) couple to one another with an arbitrary coupling (the electric charge in QED's case). So Fermi, by analogy, wrote down all the terms he could think of containing the currently known fields, hoping that one of these extra terms in the equation could describe the weak interaction.

 

One such term involved 4 fermions. However, since the Largangian has mass dimension 4 (since there are 4 space-time dimensions) but fermion fields are dimension 3/2 each, a term with 4-fermion fields in it has dimension 6, so he had to divide the term by some arbitrary energy squared (in other words, his coupling constant G_F in front of the term had dimension 1/mass²).

 

Surprisingly this theory worked quite well, and with this simple addition of one term to the Lagrangian he could explain lots of physics. Unfortunatetly, when one does quantum corrections with this model, infinities crop up all over the place. You basically can't make any accurate predictions; you have to stop at tree-level (the semi-clasical level) and be happy with getting your predictions almost right.

 

The reason for this is the mass dimension of the extra term. Later, when Pati, Salam and Weinberg came up with the SU(2) theory, they had W and Z bosons in them and all the terms in the Lagrangian, which contain these new fields and the fermions have mass dimension 4. The theory behaves itself under quantum corrections and can make precise predictions. (It is renormalizable.)

 

However, if you intergrate out these new particles (the W and Z) from the Lagrangian (you are normally integrating over it anyway, so basically this is just saying that you should keep the W and Z bosons away from the initial and final states) and take the low energy limit (where Fermi's theory was useful) you reproduce Fermi's 4-fermion theory. In fact the G_F turns out to be [math]\sqrt{2}g^2/8M_W^2[/math] where M_W is the W mass and g is a coupling constant.

 

The reason Fermi's theory gave infinities, is that when you include quantum corrections you must include particles in loops at arbitrarily high energies. So even the heavy W and Z become important. He was missing their contributions and his calculation didn't work.

 

Similarly today we have a theory of gravity which is non-renormalizable. It does fine at the semi-classical level but screws up the quantum corrections. However, I am fairly sure that we are simply missing some dynamics at high energies (just like we were missing the W and Z). If we include the right stuff, I am sure the theory will become well behaved.

 

The question is, what is the 'right stuff'....?