Dirac's Relativistic Equation and the Mysterious Nature of Gravity

[actionintegral]

Teacher said "Stupid Question"

We were learning about Dirac's relativistic equation. The teacher wrote on the
board:
Free Particle Dirac Equation:
$$[ \alpha \cdot \bold {p}c + \beta mc^2]\psi = -\frac{\hbar}{i}\frac{\partial\psi}{\partial t}$$

Then the teacher said "if we want to add a potential V we just write:"
$$[ \alpha \cdot \bold {p}c + \beta mc^2 + \bold{V}]\psi = -\frac{\hbar}{i}\frac{\partial \psi}{\partial t}$$

My hand shot up and I said "what if $$V = \frac{-GMm}{x}$$" would that represent gravity?"

Teacher set his chalk down, smirked condescendingly, and mumbled something about "coupling" and "gravitaons" and yada yada blah blah blah.

So I turn to you, dear friends, what am I not being told about gravity that makes it ... special?

 

[Demystifier]

That would represent a weak unquantized gravity. Which, of course, is far from having a quantum theory of gravity.

 

[vanesch]

That would represent a weak unquantized gravity. Which, of course, is far from having a quantum theory of gravity.

It would even represent Newtonian gravity, which is not Lorentz-invariant, and hence screws up the lorentz invariance of the dirac equation...

 

[actionintegral]

Newtonian gravity, which is not Lorentz-invariant,

Pardon my thick-headedness while I blunder through this.

But the coulomb field is just a 1/r potential. It is only lorentz invariant when you consider the velocity-dependent part, the "magnetic field". And I learned in EM that any force exhibits a "magnetism' due to relativity.

I do not understand why coulomb field is lorentz invariant but gravity field is not because they have the same form.

 

[reilly]

actionintegral -- Your prof is wrong, yours is a very perceptive question. And, yes, you could toss in the gravitational potential , and the electrostatic potential, and a static magnetic field. But, note that when so doing, you give up any rights to a covariant formulation of your problem or system. Rather, you are doing single particle dynamics of a relativistic electron in a classical world, in a particular inertial frame. and we know that this approach often works quite well.

The issue of bound states in QFT is just a bit difficult -- both mathematically and conceptually -- check out Bethe-Salpeter eq, for example.

Regards,
Reilly Atkinson

 

[actionintegral]

single particle dynamics in a classical world, and we know that this approach often works quite well.

Regards,
Reilly Atkinson

So even in QED, when we use a classical hamiltonian for the $$(\frac {\phi}{c}, A)$$ four-potential, we are "mixing" classical physics and quantum physics?

 

[reilly]

When you use quantized fields, it's a totally diferent ball game. We tend to mix CM and QM a lot. For example we know classically that gravity is not very important in most elementary particle experiments, but we don't know from in the quantum world. So we use classical to neglect gravity in most QM work. Not to worry.Regards,
Reilly Atkinson

 

[selfAdjoint]

So even in QED, when we use a classical hamiltonian for the $$(\frac {\phi}{c}, A)$$ four-potential, we are "mixing" classical physics and quantum physics?

It has long troubled physicists, but they don't seem to be able to avoid it, that to develop a quantum theory you must start with a classical theory (which is "known to be wrong") and then quantize it. In the case of QED most developments start from a classical Lagrangian rather than a Hamiltonian, but the principle is the same.

 

[actionintegral]

Ok - so gravity is weak compared to EM. But all I ever hear about is how "gravity cannot be quantized". The independent forum is full of people
trying to quantize gravity.

My question is really simple - where precisely does the controversy arise so that I can learn the difference between gravity the coulomb field?

 

[reilly]

It's the difference between special and general relativity; between spin one and spin two, between more-or-less linear and more-or-less nonlinear. Google will give you more than you care to know about the subject.
Regards,
Reilly Atkinson

 

[actionintegral]

It's the difference between special and general relativity; between spin one and spin two, beteen more-or-less linear and more-or-less nonlinear. Google will give you more than you care to know about the subject.
Regards,
Reilly Atkinson

Yes but google is like taking a bath in a tub full of wiki. That's why I use this forum so that I can get some direct answers. But I will do as you suggest. I will read up on "quantization of gravity" and search for the unresolved issues.

 

[Dense]

I thought this was helpful.

http://72.14.209.104/search?q=cache...r+That+Matter)?"+pdf&hl=en&gl=us&ct=clnk&cd=1

 

[Demystifier]

It's the difference between special and general relativity; between spin one and spin two, beteen more-or-less linear and more-or-less nonlinear.

Let me put the differences in a more logical order:
The gravitational force between two gravitational "charges" of the same sign (we call them masses) is attractive, while that for the electromagnetic case is repulsive.
It then turns out that such attractive force cannot be consistently achieved with a spin-1 field.
Assuming that the force is described by a spin-2 field, the Einstein equation necessarily emerges, which, unfortunately, is nonlinear.
The Einstein equation simplifies if it is (re)interpreted in terms of curved spacetime, but it remains nonlinear.

 

[Epicurus]

You will get onto coupling electromagnetism with the dirac equation soon, hopefully showing how it comes about through gauge invariance. In this instance, the strength of the coupling is related to charge, e. The problem of having a gauge invariant theory for gravitation, where the coupling strength, or charge is the gravitational mass. The equivalence principle states that gravitational mass is equal to the inertial mass. The inertial mass is already present in the free field equation and so the gauge transformation becomes a function of the coupling constant and mass becomes a function of position.