Dirac Equation: Combining Quantum Mechanics and Special Relativity

[cube137]

When Dirac tried to combine Quantum Mechanics and Special Relativity. Wasn't he initially worried that one was undeterministic (QM) and the second was continuous (SR). They are supposed to be incompatible. yet he combined them. Did Dirac do it by just considering the time dilation and other relativistic components like Lorentz transformation? I thought the essense of Einstein Special Relativity was the worldlines.. but you know quantum particles don't have worldlines.

This also means SR worldlines are not the main aspect of relativity and the concept can be discarded, isn't it. You can make equations that combine QM and SR without taking it into account. What part of SR is the most important then creating QFT? We newbies are only exposed to the relativity geometry and worldlines so we need to ask this.

I hope the mentors can give others chance to answer this and not just lock threads without mercy (this can leave many questions left unanswered and make newbies nights sleepless). Ty

 

[PeterDonis]

Did Dirac do it by just considering the time dilation and other relativistic components like Lorentz transformation?

Do you have a specific reference of Dirac's? What does it say?

I thought the essense of Einstein Special Relativity was the worldlines

Why do you think that?

quantum particles don't have worldlines

Classical SR does not model quantum particles.

What part of SR is the most important then creating QFT?

Lorentz invariance.

We newbies are only exposed to the relativity geometry and worldlines

Sure, if you read textbooks on classical SR. But what if you read textbooks on QFT?

 

[PeterDonis]

I hope the mentors can give others chance to answer this and not just lock threads without mercy

As a separate note, it's never a good idea to ask for your thread not to be locked, particularly on the grounds that you need an answer to a question. Not all questions that people think they need answers to fall within the PF rules. That's just the way it is.

 

[cube137]

Do you have a specific reference of Dirac's? What does it say?

https://en.wikipedia.org/wiki/Dirac_equation
The part about "Making the Schrödinger equation relativistic":
"... The left side represents the square of the momentum operator divided by twice the mass, which is the non-relativistic kinetic energy. Because relativity treats space and time as a whole, a relativistic generalization of this equation requires that space and time derivatives must enter symmetrically as they do in the Maxwell equations that govern the behavior of light — the equations must be differentially of the same order in space and time. In relativity, the momentum and the energies are the space and time parts of a spacetime vector, the four-momentum, and they are related by the relativistically invariant relation..."

Why do you think that?

We newbies were first exposed to the geometric side of SR and GR. Without the worldlines, there is no geodesics and no spacetime curvature. So I thought worldlines were the core of SR and GR. But it seems you can do away with them like in the Dirac Equation. In GR then, what is the core essence of it if it's not worldlines and curvature? In SR it's Lorentz Invariance. So the geometry part can indeed be discarded since Dirac Equation has quantum particles that don't have worldlines?

Classical SR does not model quantum particles.

I know. Yet Dirac Equation or Quantum Field Theory in general can combine quantum particles and classical SR.. how did they discard worldlines and yet use the essence of SR? If it's Lorentz Invariance.. how do you discard the worldlines and geodesics and spacetime curvature in GR and still use the essence of GR?

Lorentz invariance.

Sure, if you read textbooks on classical SR. But what if you read textbooks on QFT?

See above.

Hope others can share their views too before all this important stuff gets unaccessible and unanswerable again. Thanks!

 

[PeterDonis]

Without the worldlines, there is no geodesics and no spacetime curvature.

Sure there are. Geodesics and curvature do not require individual particles to have particular worldlines. They are geometric properties of spacetime itself.

If it's Lorentz Invariance

Do you understand what Lorentz invariance is? Do you understand how the Schrödinger Equation is not Lorentz invariant whereas the Dirac equation is?

the geometry part can indeed be discarded

No, it can't. Discarding worldlines for individual particles does not mean discarding geometry. See above.

 

[PeterDonis]

Hope others can share their views too before all this important stuff gets unaccessible and unanswerable again.

Please read my post #3. If you make any further statements along these lines you will receive a warning and this thread will be closed.

 

[Nugatory]

I know. Yet Dirac Equation or Quantum Field Theory in general can combine quantum particles and classical SR.. how did they discard worldlines and yet use the essence of SR? If it's Lorentz Invariance.. how do you discard the worldlines and geodesics and spacetime curvature in GR and still use the essence of GR?

They didn't. The "essence of SR" is a flat, no curvature, spacetime and you don't need any GR to work with it.

However, I don't think there is any B-level answer to the question of how quantum mechanics works with special relativity. The best I can come up with is: We have a Lorentz-invariant theory that allows us to calculate the probability of finding our particle at any given point in spacetime (not space!) if we happen to have a detector there.

 

[cube137]

Sure there are. Geodesics and curvature do not require individual particles to have particular worldlines. They are geometric properties of spacetime itself.

Do you understand what Lorentz invariance is? Do you understand how the Schrödinger Equation is not Lorentz invariant whereas the Dirac equation is?

Yes, Schrödinger Equation uses Newtonian space and time while Dirac equation uses the Minkowski spacetime.

No, it can't. Discarding worldlines for individual particles does not mean discarding geometry. See above.

What topics in SR and GR does this topic fall where I can find the statements "
Geodesics and curvature do not require individual particles to have particular worldlines. They are geometric properties of spacetime itself"? Actually yesterday I tried to find the answer and this is the path I took (or materials I used to research). I went to wiki and saw the following:

And I went to library and tried to reread what I read decade ago that strings where to supposed to handle it. And found the following:

"Einstein's general relativity says no, the fabric of space cannot tear. The equations of general relativity are firmly rooted in Riemannian geometry and, as we noted in the preceding chapter, this is a framework that analyzes distortions in the distance relations between nearby locations in space. In order to speak meaningfully about these distance relations, the underlying mathematical formalism requires that the substrate of space is smooth—a term with a technical mathematical meaning, but whose everyday usage captures its essence: no creases, no punctures, no separate pieces "stuck" together, and no tears. Were the fabric of space to develop such irregularities, the equations of general relativity would break down, signaling some or other variety of cosmic catastrophe—a disastrous outcome that our apparently well-behaved universe avoids.
...
"A week or so after I arrived, Witten and I were chatting in the Institute's courtyard, and he asked about my research plans. I told
him about the space-tearing flops and the strategy we were planning to pursue. He lit up upon hearing the ideas, but cautioned that he thought the calculations would be horrendously difficult. He also pointed out a potential weak link in the strategy I described, having to do with some work I had done a few years earlier with Vafa and Warner. The issue be raised turned out to be only tangential to our approach for understanding flops, but it started him thinking about what ultimately turned out to be related and complementary issues.

Aspinwall, Morrison, and I decided to split our calculation in two pieces. At first a natural division might have seemed to involve first extracting the physics associated with the final Calabi-Yau shape from the upper row of Figure 11.5, and then doing the same for the final Calabi-Yau shape from the lower row of Figure 11.5. If the mirror relationship is not shattered by the tear in the upper Calabi-Yau, then these two final Calabi-Yau shapes should yield identical physics, just like the two initial Calabi-Yau shapes from which they evolved. (This way of phrasing the problem avoids doing any of the very difficult calculations involving the upper Calabi-Yau shape just when it tears.) It turns out, though, that calculating the physics associated with the final Calabi-Yau shape in
the upper row is pretty straightforward. The real difficulty in carrying out this program lies in first figuring out the precise shape of the final Calabi-Yau space in the lower row of Figure 11.5—the putative mirror of the upper Calabi-Yau—and then in extracting
the associated physics."

Is this related to it? That strings where supposed to replace the particles in worldlines by becoming worldsheets? But then Dirac Equation doesn't have strings and I'm just sharing the stuff I'm reading to answer it. What reference can you give that can answer my question as what you said at start :
"Geodesics and curvature do not require individual particles to have particular worldlines. They are geometric properties of spacetime itself."

 

[PeterDonis]

Schrödinger Equation uses Newtonian space and time while Dirac equation uses the Minkowski spacetime

Yes. Now where in what you said here is there any mention of worldlines?

What topics in SR and GR does this topic fall where I can find the statements "
Geodesics and curvature do not require individual particles to have particular worldlines

I doubt you will find this as an explicit statement in any textbook or peer-reviewed paper on SR or GR. That's because it's considered too obvious to need an explicit statement. A particular spacetime, curved or not, is a geometric object in and of itself. Given that geometric object, its geodesics are determined. All of that can be done without associating any curves with particular particles. Just as you can define a 2-sphere (like the surface of an idealized non-rotating planet) as a geometric object, and determine its geodesics (the great circles), without ever having to associate any curves with particular objects. This is basic geometry.

I went to wiki

You should be looking at textbooks or peer-reviewed papers, not Wikipedia.

I went to library and tried to reread what I read decade ago that strings where to supposed to handle it

String theory is a whole other can of worms that I would advise you to not even think about opening at this stage of your knowledge. All it will do is confuse you further.

 

[cube137]

Yes. Now where in what you said here is there any mention of worldlines?
I doubt you will find this as an explicit statement in any textbook or peer-reviewed paper on SR or GR. That's because it's considered too obvious to need an explicit statement. A particular spacetime, curved or not, is a geometric object in and of itself. Given that geometric object, its geodesics are determined. All of that can be done without associating any curves with particular particles. Just as you can define a 2-sphere (like the surface of an idealized non-rotating planet) as a geometric object, and determine its geodesics (the great circles), without ever having to associate any curves with particular objects. This is basic geometry.

But without knowing individual worldlines.. how do you know if two test particles were in parallel geodesic, converging or diverging geodetic? This is the example we are often exposed as newbies so it's deeply ingrained in our mind that Spacetime is about how these objects behave.

You should be looking at textbooks or peer-reviewed papers, not Wikipedia.
String theory is a whole other can of worms that I would advise you to not even think about opening at this stage of your knowledge. All it will do is confuse you further.

 

[bhobba]

It much simpler than you are trying to make it.

What worried Dirac was Einsteins famous equation E^2 = p^2 + m^2. (units c=1)

You write that using the quantum rules and you get the famous Klein-Gordon equation. So far so good. But it had problems - such as nonsense like negative probabilities. The answer was not known at the time - it simply meant rather than probabilities going negative you got positive probabilities of antiparticles - but that understanding came later.

Take a look at the Schrödinger equation - it was the quantum version of E = p^2/2m.

Notice something - we have E instead of E^2. So maybe that's the culprit - so write Einsteins equation as E = sqrt (p^2 + m^2). Now in QM p and E are operators. How do you take the square root of an operator? Dirac, being the smart guy he was figured out a tricky way of doing that:
https://en.wikipedia.org/wiki/Dirac_equation

See the section on Diracs Coup..

It was a master stroke. It did away with negative probabilities but it had this damnable negative energy. This led to the hole theory. But since then it has been superseded by Quantum Field Theory that solved both issues - negative probabilities and energies were really manifestations of antiparticles.

Also, just out of interest see the following paper:
http://cds.cern.ch/record/944002/files/0604169.pdf

By setting m=0 the square-root is much easier to take - if the wave function has no sources (which mathematically is ∇.ψ = 0) its simply ∇Xψ which gives Maxwell's equations.

So by being a bit sneaky you can get a lot out of that seemingly simple Klein Gordon equation.

Thanks
Bill

 

[PeterDonis]

without knowing individual worldlines.. how do you know if two test particles were in parallel geodesic, converging or diverging geodetic?

As I've already said, you can determine the geodesics of a geometry even if there are no actual objects that are assigned particular curves. The geodesics are properties of the geometry, not properties of individual objects.

Whether or not a particular object happens to have a geodesic as its worldline is a separate question, which only makes sense if you are working in a framework (classical GR/SR) in which individual objects have worldlines to begin with. So you're just confusing yourself by asking a question that doesn't apply to the case (quantum field theory) you're considering.