Understanding Extra Internal Degree of Freedom from QM and SR

[cefarix]

I've been trying to understand something for the past few days but can't seem to get my head around it.

Could someone please explain to me how an extra internal degree of freedom arises for a particle when special relativity is combined with quantum mechanics?

 

[Meir Achuz]

You have to read about the Dirac equation. There is a lot of math involved to get the spin out.

 

[meopemuk]

See Wigner's theory of unitary irreducible representations of the Poincare group, e.g., in
S. Weinberg "The quantum theory of fields" vol.1 chapter 2.

 

[JustinLevy]

You have to read about the Dirac equation. There is a lot of math involved to get the spin out.

May I ask a follow up question?
I am still working up to learning quantum field theory, but was told at one point that the Dirac equation, a relativistic formulation of quantum mechanics for spin 1/2 particles, was a stepping stone from non-relativistic quantum mechanics (Schrödinger's equation) to a full relativistic quantum field theory.

I'm sure this will make more sense in hind-sight, but if you could give a hint of what is to come if I learn QFT, what was the fundamental 'insight' that led to the final step? What made it necessary? Is there any experimental predictions that the Dirac equations differs from quantum field theory? What can I use the Dirac equation for, and what should I avoid? (Since an 8x8 matrix and 8 component vector can also satisfy the Dirac equations, does this mean I could use it for a hypothetical 3/2 spin particle as well? How do we know what particles it applies to?)

For example when the 'negative energy' solutions come up, most books and lecture notes I've seen say something to the effect of handwaving it away and saying 'this will be made more explicit in quantum field theory'.

 

[meopemuk]

I'm sure this will make more sense in hind-sight, but if you could give a hint of what is to come if I learn QFT, what was the fundamental 'insight' that led to the final step? What made it necessary?

The fundamental difference between ordinary quantum mechanics and QFT is that the former usually deals with systems having fixed number of particles, while the latter permits also treatment of processes in which particles are created or annihilated. QM is formulated in Hilbert spaces with (fixed) N particles, while the number of particles is not fixed in the Fock space of QFT. Quantum states with any N from 0 to infinity are allowed in the Fock space.

 

[JustinLevy]

Then for situations with a fixed number of particles, the Dirac equation is equivalent? Or only in some specific limit?

And how would considering states with varying numbers of particles give a different insight on the 'negative energy' states?

Sorry for all the questions. If someone could suggest a good QFT book that would be great too. I could follow the dirac equation and klien-gordon equation, but when I started reading about QFT the book I had (don't remember at the moment, I can look it up if it helps) jumped straight into everything and I couldn't even tell what observables or a measurement were anymore.

 

[meopemuk]

Then for situations with a fixed number of particles, the Dirac equation is equivalent? Or only in some specific limit?

In my opinion, the usefulness of the Dirac equation is grossly overstated in most textbooks. It is true that one can calculate rather accurately the fine structure of the hydrogen atom in this approach. However, there is an alternative approach based on the Breit Hamiltonian, which is equally accurate, and which has a more solid theoretical foundation, in my opinion. Both these theories are approximate. Both of them fail to describe the Lamb shifts of atomic levels, for which the full-fledged QFT is needed.

And how would considering states with varying numbers of particles give a different insight on the 'negative energy' states?

The 'negative energy' states were fun to discuss in 1920's-1930's. But I don't see the point (rather than historical) to mention them in modern QFT textbooks. They are just confusing and not helpful. All physical states have non-negative energies.

Sorry for all the questions. If someone could suggest a good QFT book that would be great too. I could follow the dirac equation and klien-gordon equation, but when I started reading about QFT the book I had (don't remember at the moment, I can look it up if it helps) jumped straight into everything and I couldn't even tell what observables or a measurement were anymore.

My favorite is Weinberg's "The quantum theory of fields". Some people say that this book is difficult to read. But IMHO it is the only major textbook which presents correctly the logic and motivation for introducing quantum fields.

 

[JustinLevy]

My favorite is Weinberg's "The quantum theory of fields". Some people say that this book is difficult to read. But IMHO it is the only major textbook which presents correctly the logic and motivation for introducing quantum fields.

Thanks for the suggestion!
My problem when I previously tried to read and learn QFT is that I didn't understand what anything meant physically and very quickly got lost. I didn't really understand what an observable was anymore, nor how to represent a measurement, nor what to interpret the 'field' as (in what sense or limit can I relate it back to the wavefunction interpretation?). Does the book go over stuff like this in the lead up and motivation? If so, I'm going to go to amazon.com and order this book right now!

 

[cefarix]

An electron's spin causes it to have an associated magnetic moment. What is the orientation of the poles of the moment?

 

[Meir Achuz]

An electron's spin causes it to have an associated magnetic moment. What is the orientation of the poles of the moment?

The magnetic moment of an electron is a point magnetic moment, which does not have "poles" in the classical sense. The magnetic moment vector is aligned with the spin vector, with
[\tex]{\vec\mu}=-e{\vec s}/min natural units.

 

[Meir Achuz]

May I ask a follow up question?
I am still working up to learning quantum field theory, but was told at one point that the Dirac equation, a relativistic formulation of quantum mechanics for spin 1/2 particles, was a stepping stone from non-relativistic quantum mechanics (Schrödinger's equation) to a full relativistic quantum field theory.

I'm sure this will make more sense in hind-sight, but if you could give a hint of what is to come if I learn QFT, what was the fundamental 'insight' that led to the final step? What made it necessary? Is there any experimental predictions that the Dirac equations differs from quantum field theory? What can I use the Dirac equation for, and what should I avoid? (Since an 8x8 matrix and 8 component vector can also satisfy the Dirac equations, does this mean I could use it for a hypothetical 3/2 spin particle as well? How do we know what particles it applies to?)

For example when the 'negative energy' solutions come up, most books and lecture notes I've seen say something to the effect of handwaving it away and saying 'this will be made more explicit in quantum field theory'.

You ask many question, and will get a variety of answers on the forum. I suggest you take a good course in QM and then in QFT. This would answer all your questions in the proper order, rather than a hodge podge. One suggestion I do have is that you read the best QM textbook ever, which is the original one by Dirac. It will give you the original, and still the most compelling, motivation for going from NRQM to RQM, and then why RQM cannot be simple one particle QM.

 

[koolmodee]

I've been trying to understand something for the past few days but can't seem to get my head around it.

Could someone please explain to me how an extra internal degree of freedom arises for a particle when special relativity is combined with quantum mechanics?

What do you mean here? What extra internal degree of freedom?

EDIT: Don't forget there is also the Gordon-Klein equation that we get from combing QM and SR.

The desire to describe the electron relativistically, a spin 1/2 particle, led Dirac to his equation.

Also, in QM we also can handle spin, nonrelativistcally of course.

 

[koolmodee]

Then for situations with a fixed number of particles, the Dirac equation is equivalent? Or only in some specific limit?

And how would considering states with varying numbers of particles give a different insight on the 'negative energy' states?

Sorry for all the questions. If someone could suggest a good QFT book that would be great too. I could follow the dirac equation and klien-gordon equation, but when I started reading about QFT the book I had (don't remember at the moment, I can look it up if it helps) jumped straight into everything and I couldn't even tell what observables or a measurement were anymore.

Everybody note there is a first quantization and a second quantization.

First quantization is not the real deal, only gives less correct answers, particle number is fixed and it suffers from negative energy solutions.First quantization gives relativistic wave equations. They are relativistic versions of the Schrödinger equation if you will. Klein Gordon for spin zero, Dirac for half spin.

In QFT, we quantize relativistic operator-valued fields, and than use the relativistic equations of motions of Klein-Gordon and Dirac to describe how these quantum fields propagate.
These quantum fields are built out of raising and lower operators, so they handle fine the varying number of particle process, by hitting on vacuum states.

 

[meopemuk]

Thanks for the suggestion!
My problem when I previously tried to read and learn QFT is that I didn't understand what anything meant physically and very quickly got lost. I didn't really understand what an observable was anymore, nor how to represent a measurement, nor what to interpret the 'field' as (in what sense or limit can I relate it back to the wavefunction interpretation?). Does the book go over stuff like this in the lead up and motivation? If so, I'm going to go to amazon.com and order this book right now!

To my taste, Weinberg's book does not have sufficient foundational discussions of this sort. Nevertheless, this book is a good investment of your money if you seriously want to learn QFT. Regarding physical meaning and interpretation of quantum fields and their relationship to wave functions you may try http://www.arxiv.org/abs/physics/0504062