Lab Notes for a Scientific Revolution (Physics)

December 26, 2008

Finite Amplitudes Without +i\epsilon

To all,

I have now completed a paper at the link below, which summarizes the work I have been doing for the past two months (and in a deeper sense, for much of my adult life) to lay a foundation for understanding and calculating particle masses:

finite-amplitudes-without-i-epsilon

I have also taken the plunge and submitted this for peer review. ;-)?

The abstract is as follows:

By carefully reviewing how the invariant amplitude M is arrived at in the simplest Yang-Mills gauge group SU(2), we show how to arrive at a finite, pole-free amplitudes without having to resort to the “+i\epsilon prescription.”  We first review how gauge boson mass is generated in the SU(2) action via spontaneous symmetry breaking in the standard model, and then carefully consider the formation of finite, on-shell amplitudes, without +i\epsilon .

Comments are welcome, and I wish everyone a happy holiday and New Year!

Jay.

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December 11, 2008

Understanding the QCD Meson Mass Spectrum

Dear Friends:

It has been awhile since I last posted and it is good to be back.

Almost two years ago in the course of my work on Yang Mills, I came across what I believe is an approach by which mass spectrum of the massive mesons of QCD might be understood.  I had what I still believe is the right concept, and many of the pieces, but I could not figure out the right execution of the concept in complete detail.  Over the past year and a half I walked away from this to let the dust settle and to also arrive at a place where the basic principles of quantum field theory were no longer “new” to me but had become somewhat ingrained.  Now, I believe I have found the right way to execute this concept, and the results are intriguing.

In the file linked below, which I will update on a regular basis in the coming days:

https://jayryablon.files.wordpress.com/2008/12/su-3-paper-20.pdf,

 I review how mass is known to be generated in SU(2), as a template for considering SU(3) QCD.  I have tried to explain as simply as possible, what I believe to be the origin of QCD meson masses, as well as to lay the foundation for theoretically predicting these.  Keep in mind, finding out how the vector mesons of QCD obtain their non-zero masses, which make the QCD interaction short range despite supposedly-massless gluons, is one aspect of the so-called “mass gap” problem, see point 1) on page 3 of
http://www.claymath.org/millennium/Yang-Mills_Theory/yangmills.pdf at
http://www.claymath.org/millennium/Yang-Mills_Theory/.

Then, I extend this development, in detail, to SU(3).

Several interesting results are already here:

1)  This approach neatly solves the problem of propagator poles (infinities) in a manner which I believe has not heretofore been discovered.  Goodbye to the +i\eta prescription, off mass-shell particles, and other inelegant dodges to achieve a finite propagator.

2) This approach may solve the confinement and the mass gap problems simultaneously.  It is important to understand that electroweak SU(2)xU(1) is a special case in which the gauge bosons are synonymous with the observed vector mesons, but that in SU(3) and higher order theories they are not.  The gauge bosons aka gluons, which show up in the Lagrangian, are not observed.  What is observed are the vector mesons which pass through to the denominator of the propagator in the invariant amplitude.

3) There emerges is a quantum number that is restricted to three discrete values, and depending on which value of chosen, all the meson masses are scaled up or down on a wholesale basis.  I believe that this may resolve the problem of generation replication.

I expect to be churning out mass calculations in the next day or two.  You may wish to check out the meson mass tables at http://pdg.lbl.gov/2008/tables/rpp2008-qtab-mesons.pdf, because that table contains the data which I am going to try to fit to equation (6.1), via (6.5).

Hope you enjoy!

Jay.

June 19, 2008

A New Lab Note: Commutation of Linear Rest Mass with Canonical Position

It has been awhile since my last blog entry, but if you want to check out some my recent wanderings through physicsland, check out sci.physics.foundations, relativity, and research.

Here, I would like to show a rather simple calculation, which may cast a different light on how one needs to think about the canonical commutation relationship \left[x_{j} ,p_{k} \right]=i\eta _{jk} ;\; j,k=1,2,3.  I would very much like your comments in helping me sort this through.  You may download this in pdf form at https://jayryablon.files.wordpress.com/2008/06/linear-mass-commutator-calculation.pdf.

I.  A Known Square Mass Commutation Calculation

 Consider a particle of mass m as a single particle system.  Consider canonical coordinates x_{\mu } , and that at least the space coordinates x_{j} ;\; j=1,2,3 are operators.  If we require that the mass m must commute with all operators, then we must have \left[x_{\mu } ,m\right]=0, and by easy extension, \left[x_{\mu } ,m^{2} \right]=0.  It is well known that the commutation condition \left[x_{\mu } ,m^{2} \right]=0, taken together with the on-shell mass relationshipm^{2} =p^{\sigma } p_{\sigma } and the single-particle canonical commutation relationship \left[x_{j} ,p_{k} \right]=i\eta _{jk} ;\; j,k=1,2,3, where diag\left(\eta _{\mu \nu } \right)=\left(-1,+1,+1,+1\right) is the Minkowski tensor, leads inexorably to the commutation relationship:

\left[x_{k} ,p_{0} \right]=-ip_{k} /p^{0} =-iv_{k}    (1.1)

where v_{k} is the particle velocity (in c=1 units) along the kth coordinate.  I leave the detailed calculation as an exercise for the reader not familiar with this calculation, and refer also to the sci.physics.research thread at http://www.physicsforums.com/archive/index.php/t-142092.html or http://groups.google.com/group/sci.physics.research/browse_frm/thread/d78cbfecf703ff6a.

 I would ask for your comments on the following calculation, which is totally analogous to the calculation that leads to (1.1), but which is done using the linear mass m rather than the square mass m^{2} , and using the Dirac equation written as m\psi =\gamma ^{\nu } p_{\nu } \psi , in lieu of what is, in essence, the Klein Gordon equation m^{2} \phi =p^{\sigma } p_{\sigma } \phi that leads to (1.1).

2.  Maybe New?? Linear Mass Commutation Calculation

 Start with Dirac’s equation written as:

m\psi =\gamma ^{\nu } p_{\nu } \psi .  (2.1) 

Require that:

\left[x_{\mu } ,m\right]=0   (2.2)

 Continue to use the canonical commutator \left[x_{j} ,p_{k} \right]=ig_{jk} .  Multiply (2.1) from the left by x_{\mu } noting that \left[\gamma ^{\nu } ,x_{\mu } \right]=0 to write:

x_{\mu } m\psi =\gamma ^{\nu } x_{\mu } p_{\nu } \psi =\gamma ^{0} x_{\mu } p_{0} \psi +\gamma ^{j} x_{\mu } p_{j} \psi .  (2.3) 

This separates into:

 \left\{\begin{array}{c} {x_{0} m\psi =\gamma ^{0} x_{0} p_{0} \psi +\gamma ^{j} x_{0} p_{j} \psi } \\ {x_{k} m\psi =\gamma ^{0} x_{k} p_{0} \psi +\gamma ^{j} x_{k} p_{j} \psi } \end{array}\right. .  (2.4)

  Now, use the canonical relation \left[x_{j} ,p_{k} \right]=i\eta _{jk} to commute the space (k) equation, thus:

 \begin{array}{l} {x_{k} m\psi =\gamma ^{0} x_{k} p_{0} \psi +\gamma ^{j} x_{k} p_{j} \psi =\gamma ^{0} x_{k} p_{0} \psi +\gamma ^{j} \left(p_{j} x_{k} +i\eta _{jk} \right)\, \psi } \\ {=\gamma ^{0} x_{k} p_{0} \psi +\gamma ^{j} p_{j} x_{k} \psi +i\gamma _{k} \psi } \\ {=\gamma ^{0} x_{k} p_{0} \psi +mx_{k} \psi -\gamma ^{0} p_{0} x_{k} \psi +i\gamma _{k} \psi } \end{array} .  (2.5)

In the final line, we use Dirac’s equation written as mx_{\mu } \psi =\gamma ^{\nu } p_{\nu } x_{\mu } \psi =\gamma ^{0} p_{0} x_{\mu } \psi +\gamma ^{j} p_{j} x_{\mu } \psi , and specifically, the \mu =k component equation \gamma ^{j} p_{j} x_{k} \psi =mx_{k} \psi -\gamma ^{0} p_{0} x_{k} \psi .

 If we require that \left[x_{\mu } ,m\right]=0, which is (2.2), then (2.5) reduces easily to:

 \gamma ^{0} \left[x_{k} ,p_{0} \right]\psi =-i\gamma _{k} \psi ,  (2.6)

Finally, multiply from the left by \gamma ^{0} , and employ \gamma ^{0} \gamma _{k} \equiv \alpha _{k} and \gamma ^{0} \gamma ^{0} =1 to write:

\left[x_{k} ,p_{0} \right]\, \psi =-i\alpha _{k} \psi .  (2.7) 

If we contrast (2.7) to (1.1) written as \left[x_{k} ,p_{0} \right]\phi =-iv_{k} \phi , we see that the velocity p_{k} /p^{0} =v_{k} has been replaced by the Dirac operator \alpha _{k} , that is, v_{k} \to \alpha _{k} .

3.  Questions

 Here are my first set of questions:

 1)  Is the calculation leading to (2.7) correct, and is (2.7) a correct result, or have I missed something along the way?

2)  If (2.7) is correct, has anyone seen this result before?  If so where?

3)  Now use the plane wave \psi =ue^{ip^{\sigma } x_{\sigma } } so that we can work with the Dirac spinors u\left(p^{\mu } \right), and rewrite (2.7) as:

\left\{\begin{array}{c} {\left(\alpha _{k} -\lambda \right)\, u=0} \\ {\lambda =i\left[x_{k} ,p_{0} \right]} \end{array}\right.  

The upper member of (3.1) is an eigenvalue equation.  Reading out this equation, I would say that the commutators \lambda =i\left[x_{k} ,p_{0} \right] are the eigenvalues of the Dirac \alpha _{k} matrices, which are:

{\bf \alpha }=\left(\begin{array}{cc} {0} & {{\bf \sigma }} \\ {{\bf \sigma }} & {0} \end{array}\right) and {\bf \alpha }=\left(\begin{array}{cc} {-{\bf \sigma }} & {0} \\ {0} & {{\bf \sigma }} \end{array}\right) ,  (3.2)

in the respective Pauli/Dirac and Weyl representations, and that the u are the eigenvectors associated with these eigenvalues \lambda =i\left[x_{k} ,p_{0} \right].  Am I wrong?  If not, how would one interpret this result?  Maybe the commutators \left[x_{j} ,p_{k} \right]=i\eta _{jk} can be discussed in the abstract, but it seems to me that the commutators \lambda =i\left[x_{k} ,p_{0} \right] can only be discussed as the eigenvalues of the matrices \alpha _{k} with respect to the eigenstate vectors u.  This, it seems, would put canonical commutation into a somewhat different perspective than is usual.

Just as Dirac’s equation reveals some features that cannot be seen strictly from the Klein Gordon equation, the calculation here seems to reveal some features about the canonical commutators that the usual calculation based on \left[x_{\mu } ,m^{2} \right]=0 and m^{2} =p^{\sigma } p_{\sigma } cannot, by itself, reveal.

I’d appreciate your thoughts on this, before I proceed downstream from here.

Thanks,

Jay.

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