Lab Notes for a Scientific Revolution (Physics)

April 24, 2008

Heisenberg Uncertainty and Schwinger Anomaly: Two Sides of the Same Coin?

 In section 3 of Heisenberg Uncertainty and Schwinger Anomaly: Two Sides of the Same Coin?, I have posted a calculation which shows why the Schwinger magnetic anomaly may in fact be very tightly tied to the Heisenberg inequality \Delta x\Delta p\ge {\tfrac{1}{2}} \hbar .  The bottom line result, in (3.11) and (3.12), is that the gyromagnetic “g-factor” for a charged fermion wave field with only intrinsic spin (no angular momentum) is given by:

\left|g\right|=2\frac{\left(\Delta x\Delta p\right)}{\hbar /2} \ge 2   (3.11)

It is also helpful to look at this from the standpoint of the Heisenberg principle as:

\Delta x\Delta p=\frac{\left|g\right|}{2} \frac{\hbar }{2} \ge \frac{\hbar }{2}   (3.12)

The draft paper linked above has details of how I got here.  Below, are some ways to think about this:

 First, if (3.11) is true, then the greater than or equal to inequality of Heisenberg says, in this context, that the magnitude of the intrinsic g-factor of a charged wavefunction is always greater than or equal to 2.  That is, the inequality \Delta x\Delta p\ge {\tfrac{1}{2}} \hbar becomes another way of stating a parallel inequality \left|g\right|\ge 2.  We know this to be true for the charged leptons, which have g_{e} /2=1.0011596521859, g_{\mu } /2=1.0011659203, and g_{\tau } /2=1.0011773 respectively. [The foregoing data is extracted from W.-M. Yao et al., J. Phys. G 33, 1 (2006)]

 Secondly, the fact that the charged leptons have g-factors only slightly above 2, suggests that these a) differ from perfect Gaussian wavefunctions by only a very tiny amount, b) the electron is slightly more Gaussian than the muon, and the muon slightly more-so than the tauon.  The three-quark proton, with g_{P} /2=2.7928473565, is definitively less-Gaussian the charged leptons.  But, it is intriguing that the g-factor is now seen as a precise measure of the degree to which a wavefunction differs from a perfect Gaussian.

 Third, (3.11) states that the magnetic moment anomaly via the g-factor is a precise measure of the degree to which \Delta x\Delta p exceeds \hbar /2.  This is best seen by writing (3.11) as (3.12).

Thus, for the electron, \left(\Delta x\Delta p\right)_{e} =1.0011596521859\cdot \left(\hbar /2\right), to give an exact numerical example.  For a different example, for the proton, \left(\Delta x\Delta p\right)_{P} =2.7928473565\cdot \left(\hbar /2\right).

 Fourth, as a philosophical and historical matter, one can achieve a new, deeper perspective about uncertainty.  Classically, it was long thought that one can specify position and momentum simultaneously, with precision.  To the initial consternation of many and the lasting consternation of some, it was found that even in principle, one could at best determine the standard deviations in position and momentum according to \Delta x\Delta p\ge {\tfrac{1}{2}} \hbar .  There are two aspects of this consternation:  First, that one can never have\Delta x\Delta p=0 as in classical theory.  Second, that this is merely an inequality, not an exact expression, so that even for a particle with \Delta x\Delta p\ge {\tfrac{1}{2}} \hbar , we do not know for sure what is its exact value of \Delta x\Delta p.  This latter issue is not an in-principle limitation on position and momentum measurements; it is a limitation on the present state of human knowledge.

 Now, while {\tfrac{1}{2}} \hbar is a lower bound in principle, the question remains open to the present day, whether there is a way, for a given particle, to specify the precise degree to which its \Delta x\Delta p exceeds {\tfrac{1}{2}} \hbar , and how this would be measured.  For example, one might ask, is there any particle in the real world that is a perfect Gaussian, and therefore can be located in spacetime and conjugate momentum space, down to exactly {\tfrac{1}{2}} \hbar .  Equation (3.12) above suggests that if such a particle exists, it must be a perfect Gaussian, and, that we would know it was a perfect Gaussian, if its g-factor was experimentally determined to be exactly equal to the Dirac value of 2.  Conversely, (3.12) tells us that it is the g-factor itself, which is the direct experimental indicator of the magnitude of \Delta x\Delta p for any given particle wavefunction.  The classical precision of \Delta x\Delta p=0 comes full circle, and while it will never return, there is the satisfaction of being able to replace this with the quantum  mechanical precision of (3.12), \Delta x\Delta p=\left|g\right|\hbar /4, rather than the weaker inequality of \Delta x\Delta p\ge {\tfrac{1}{2}} \hbar .

 Fifth, if (3.12) is correct, then since it is independently known from Schwinger that \frac{g}{2} =1+\frac{a}{2\pi } +\ldots , this would mean that we would have to have:

\Delta x\Delta p=\frac{\left|g\right|}{2} \frac{\hbar }{2} =\left(1+\frac{a}{2\pi } +\ldots \right)\frac{\hbar }{2}   (3.13)

Thus, from the perturbative viewpoint, the degree to which \Delta x\Delta p exceeds {\tfrac{1}{2}} \hbar would have to be a function of the running coupling strength \alpha =e^{2} /4\pi in Heaviside-Lorentz units.  As Carl Brannnen has explicitly pointed out to me, this means that a Gaussian wavepacket is by definition non-interacting; as soon as there is an interaction, one concurrently loses the exact Gaussian.

 Sixth, since deviation of the g-factor above 2 would arise from a non-Gaussian wavefunction such as \psi (x)=N\exp \left(-{\tfrac{1}{2}} Ax^{2} +Bx\right), the rise of the g-factor above 2 would have to stem from the Bx term in this non-Gaussian wavefuction.  In this regard, we note to start, that N\int \exp \left(-{\tfrac{1}{2}} Ax^{2} +Bx\right)dx= \sqrt{2\pi /A} \exp \left(B^{2} /2A\right), for a non-Gaussian wavefunction, versus N\int \exp \left(-{\tfrac{1}{2}} Ax^{2} \right)dx= \sqrt{2\pi /A} for a perfect Gaussian.

 Finally, to calculate this all out precisely, one would need to employ a calculation similar to that shown at, but for the non-Gaussian N\int \exp \left(-{\tfrac{1}{2}} Ax^{2} +Bx\right)dx= \sqrt{2\pi /A} \exp \left(B^{2} /2A\right) rather than the GaussianN\int \exp \left(-{\tfrac{1}{2}} Ax^{2} \right)dx= \sqrt{2\pi /A} , to arrive at the modified bottom line equation of this Wiki section.  That is the next calculation I plan, but this is enough, I believe, to post at this time.

April 18, 2008

Lab Note 6: Operator Decomposition of Intrinsic Spin

I’d like to lay out a nifty little mathematical calculation which allows a “decomposition” of the intrinsic spin matrices s^{i} ={\tfrac{1}{2}} \hbar \sigma ^{i} to include the position and momentum operators x^{i} , p^{i} , i=1,2,3.  To simplify matters, we will employ a Minkowski metric tensor with {\rm diag}\left(\eta _{\mu \nu } \right)=\left(-1,+1,+1,+1\right) so that raising and lowering the space indexes i=1,2,3 is simple and at will, and does not entail any sign reversal.  (This lab note is available in PDF form, with a recent update including a new section 2, at intrinsic-spin-decomposition-11.)

  We start with the general cross product for two three-vectors A and B.  Written in covariant (index) notation:

\left(A\times B\right)_{i} \equiv \varepsilon _{ijk} A^{j} B^{k} .   (1)

One can easily confirm this by taking, for example, \left(A\times B\right)_{3} \equiv A^{1} B^{2} -A^{2} B^{1} .  Now, let’s take the triple cross product \left(A\times B\right)\times C.  We can apply (1) to itself using \left(A\times B\right)^{j} \equiv \varepsilon ^{jmn} A_{m} B_{n} , to write:

\left[\left(A\times B\right)\times C\right]_{i} =\varepsilon _{ijk} \left(A\times B\right)^{j} C^{k} =\varepsilon _{ijk} \varepsilon ^{jmn} A_{m} B_{n} C^{k} .  (2)

 The fact that the crossing of A and B takes precedence over crossing with C is retained in the fact that A_{m} B_{n} sum with \varepsilon ^{jmn} , while C^{k} alone sums into \varepsilon _{ijk} .

 Let us now expand (2) for the component equation for which i=3.  The calculation is as such:

\begin{array}{l} {\left[\left(A\times B\right)\times C\right]_{3} =\varepsilon _{3jk} \varepsilon ^{jmn} A_{m} B_{n} C^{k} } \\ {=\varepsilon _{312} \varepsilon ^{123} A_{2} B_{3} C^{2} +\varepsilon _{312} \varepsilon ^{132} A_{3} B_{2} C^{2} +\varepsilon _{321} \varepsilon ^{231} A_{3} B_{1} C^{1} +\varepsilon _{321} \varepsilon ^{213} A_{1} B_{3} C^{1} } \\ {=A_{1} B_{3} C^{1} +A_{2} B_{3} C^{2} -A_{3} B_{1} C^{1} -A_{3} B_{2} C^{2} } \\ {=A_{1} B_{3} C^{1} +A_{2} B_{3} C^{2} +A_{3} B_{3} C^{3} -A_{3} B_{1} C^{1} -A_{3} B_{2} C^{2} -A_{3} B_{3} C^{3} } \\ {=A_{1} B_{3} C^{1} +A_{2} B_{3} C^{2} +A_{3} B_{3} C^{3} -A_{3} \left(B\cdot C\right)} \end{array},  (3)

where we have added 0=A_{3} B_{3} C^{3} -A_{3} B_{3} C^{3} to the fourth line.  Now in the final line, we hit an impasse, because B_{3} is sandwiched between the terms we would like to form into the other dot product A\cdot C.  In order to complete the calculation, we must make an assumption that the A_{i} commute with B_{3} , i.e., that \left[A_{i} ,B_{3} \right]=0.  For now, let us make this assumption.

 Therefore, we carry out the commutation in (3), and continue along to write:

\begin{array}{l} {\left[\left(A\times B\right)\times C\right]_{3} =\varepsilon _{3jk} \varepsilon ^{jmn} A_{m} B_{n} C^{k} =A_{1} B_{3} C^{1} +A_{2} B_{3} C^{2} +A_{3} B_{3} C^{3} -A_{3} \left(B\cdot C\right)} \\ {=B_{3} \left(A\cdot C\right)-A_{3} \left(B\cdot C\right)=B_{3} A_{j} C^{j} -A_{3} B_{j} C^{j} } \end{array}.  (4) 

Generalizing fully, we may now write (4) in two equivalent ways as:

\left\{\begin{array}{c} {\left(A\times B\right)\times C=-A\left(B\cdot C\right)+B\left(A\cdot C\right)\quad \quad } \\ {\varepsilon _{ijk} \varepsilon ^{jmn} A_{m} B_{n} C^{k} =-A_{i} B_{j} C^{j} +B_{i} A_{j} C^{j} } \end{array}\right. .  (5) 

The reader will observe the well-known formula for the cross product.

 Now, let’s take the cross product in which A=x, B=p and C={\bf \sigma }, where x is the position operator about the center of mass, p is the momentum operator, and {\bf \sigma } are the Pauli spin matrices.  We also take into account the Heisenberg canonical commutation relationship between the position and momentum operators, that is, \left[x_{\mu } ,p_{\nu } \right]=i\hbar \delta _{\mu \nu } .  This means that we will have to be careful at the juncture between equations (3) and (4), because the position and momentum operators along the same dimension do not commute.

 So, we return to (3) with A=x, B=p and C={\bf \sigma }, to write:

\left[\left(x\times p\right)\times {\bf \sigma }\right]_{3} =\varepsilon _{3jk} \varepsilon ^{jmn} x_{m} p_{n} \sigma ^{k} =x_{1} p_{3} \sigma ^{1} +x_{2} p_{3} \sigma ^{2} +x_{3} p_{3} \sigma ^{3} -x_{3} \left(p\cdot {\bf \sigma }\right).  (6)

To take the next step, we want to place p_{3} in front of the x_{i} .  In so doing, we can commute p_{3} with x_{i} for i=1,2.  But, for i=3, we must employ x_{3} p_{3} =p_{3} x_{3} +i\hbar .  Therefore, (6) now becomes:

 \begin{array}{l} {\left[\left(x\times p\right)\times {\bf \sigma }\right]_{3} =\varepsilon _{3jk} \varepsilon ^{jmn} x_{m} p_{n} \sigma ^{k} =p_{3} x_{1} \sigma ^{1} +p_{3} x_{2} \sigma ^{2} +\left(p_{3} x_{3} +i\hbar \right)\sigma ^{3} -x_{3} \left(p\cdot {\bf \sigma }\right)} \\ {=p_{3} \left(x\cdot {\bf \sigma }\right)-x_{3} \left(p\cdot {\bf \sigma }\right)+i\hbar \sigma _{3} =p_{3} x_{j} \sigma ^{j} -x_{3} p_{j} \sigma ^{j} +i\hbar \sigma _{3} } \end{array},  (7)

lowering the index on i\hbar \sigma ^{3} with {\rm diag}\left(\eta _{ij} \right)=\left(+1,+1,+1\right).  Now all of a sudden, i\hbar \sigma ^{3} has made an unexpected appearance.  Generalizing (7), we may write:

\left\{\begin{array}{c} {\left[\left(x\times p\right)\times {\bf \sigma }\right]=-x\left(p\cdot {\bf \sigma }\right)+p\left(x\cdot {\bf \sigma }\right)+i\hbar {\bf \sigma }\quad \quad \quad } \\ {\varepsilon _{ijk} \varepsilon ^{jmn} x_{m} p_{n} \sigma ^{k} =-x_{i} p_{j} \sigma ^{j} +p_{i} x_{j} \sigma ^{j} +i\hbar \sigma _{i} } \end{array}\right. ,  (8 )

 This is the also the well-known formula for the triple-cross product, but with an additional term i\hbar {\bf \sigma } emerging from the canonical commutation relationship.  In fact, moving terms, equation (8 ) gives us a way to decompose the intrinsic spin matrix so as to contain the position and momentum, and as we shall also see, orbital angular momentum operators.

 First, we rewrite (8 ) as:

\left\{\begin{array}{c} {i\hbar s=\left[\left(x\times p\right)\times s\right]+x\left(p\cdot s\right)-p\left(x\cdot s\right)\quad \quad \quad } \\ {i\hbar s_{i} =\varepsilon _{ijk} \varepsilon ^{jmn} x_{m} p_{n} s^{k} +x_{i} p_{j} s^{j} -p_{i} x_{j} s^{j} } \end{array}\right. ,   (9) 

where we have multiplied through by {\tfrac{1}{2}} \hbar and then set s_{i} \equiv {\tfrac{1}{2}} \hbar \sigma _{i} .  This decomposes the intrinsic spin matrix into an expression involving itself, as well as the position and momentum operators.

 Now, using the definition (1) but with A=x and B=p, let’s introduce the orbital angular momentum operator :

l^{j} \equiv \left(x\times p\right)^{j} \equiv l^{j} \equiv \varepsilon ^{jmn} x_{m} p_{n}   (10)

It is easy to see, for example, that l^{3} =x_{1} p_{2} -x_{2} p_{1} .  Using (10), we now rewrite (9) as:

\left\{\begin{array}{c} {i\hbar s=\left(l\times s\right)+x\left(p\cdot s\right)-p\left(x\cdot s\right)\quad \; \; } \\ {i\hbar s_{i} =\varepsilon _{ijk} l^{j} s^{k} +x_{i} p_{j} s^{j} -p_{i} x_{j} s^{j} } \end{array}\right. ,   (11) 

We see that part of this decomposition includes the cross-product l\times s of the orbital angular momentum with the intrinsic spin.  We may also multiply the lower equation (11) through by \varepsilon ^{mni} and then employ the commutation relationship \left[s^{m} ,s^{n} \right]=i\hbar \varepsilon ^{mni} s_{i} , to write:

\left[s^{m} ,s^{n} \right]=\varepsilon ^{mni} \varepsilon _{ijk} l^{j} s^{k} +\varepsilon ^{mni} x_{i} p_{j} s^{j} -\varepsilon ^{mni} p_{i} x_{j} s^{j} =l^{m} s^{n} +\varepsilon ^{mni} x_{i} p_{j} s^{j} -\varepsilon ^{mni} p_{i} x_{j} s^{j} .  (12) 

Note, we have also made use of \varepsilon ^{mni} \varepsilon _{ijk} =\delta ^{mni} _{ijk}

 Equation (11) allows us to decompose the total spin S for a Dirac field \psi , as follows: WORDPRESS DOES NOT LIKE THE INTEGRALS — NEED TO FIX

\left\{\begin{array}{c} {S=\int \left(\overline{\psi }s\psi \right)\, d^{3} x =-{\tfrac{i}{\hbar }} \int \left(\overline{\psi }\left[\left(l\times s\right)+x\left(p\cdot s\right)-p\left(x\cdot s\right)\right]\, \psi \right)\, d^{3} x\quad \; \;  } \\ {S_{i} =\int \left(\overline{\psi }s_{i} \psi \right)\, d^{3} x =-{\tfrac{i}{\hbar }} \int \left(\overline{\psi }\left[\varepsilon _{ijk} l^{j} s^{k} +x_{i} p_{j} s^{j} -p_{i} x_{j} s^{j} \right]\psi \right)\, d^{3} x } \end{array}\right.  (13)

See Ohanian, H., What is spin?,  equation (18).

More to follow . . .

April 14, 2008

John Archibald Wheeler RIP

Filed under: Geometrodynamics,John Archibald Wheeler,Physics,Science,Wheeler — Jay R. Yablon @ 12:19 am

I just read that John Archibald Wheeler passed away this past day. 

With Wheeler’s passing, we have lost a giant in the world of physics.  Those who have followed my work know that I have been very heavily influenced by Wheeler’s view that all of nature must have a geometrodynamic foundation.  While his dream has not yet been realized, and many have abandoned his dream in favor of other avenues, there remain a few of us quixotic die-hards who will not give up the ghost on Wheeler’s approach, because it is difficult to see how God might have done anything other than to rest nature upon geometry.  In fact, if I were to sum up in one sentence the central thrust of my research, it is to show that Wheeler was right, and that his dream of realizing a geometrodynamic foundation for all of nature has been prematurely abandoned.

Quantum field theory must certainly be credited for its astounding success in describing nature.  But, that does not mean that geometrodynamics cannot work.  It just means we have not yet been able to find out the way in which QFT and geometrodynamics are compatible and, indeed, inseparable.  One day, we will have a recognized theory of quantum geometrodynamics, and when we do, Wheeler will be recognized as the visionary who laid out the program, and who kept Einstein’s dream of a geometric foundation of nature alive for successive generations of physicists, even as many went their separate ways.

Though Wheeler was the third name on Misner, Thorne & Wheeler (and all one needs to say is MTW, and every student of physics knows exactly what book that means), he was the visionary author.  With all the encyclopedic calculation developed in MTW, it is Wheeler who wrote the “perspective” pieces, and always kept a forward eye on where physics should, and will, one day end up. 

Wheeler can teach many lessons to those who become so bogged down in physics calculation or dogma or pedagogy, that they miss or forget that the central aim of physics study is to understand how God created nature, and that the process of uncovering this understanding, fundamentally, is creative, and human.  All else is secondary. 

Farewell to one of the great souls of physics, who in this past day, has returned to his creator, and is now undoubtedly asking that creator directly, all of the questions he asked when he walked among us on this earth.  In his memory, let us rededicate ourselves to keeping alive Wheeler’s geometrodynamic program.

Another WordPress author paid his own triubute at  I am sure there will be more in the coming days from all over the scientific world.


April 13, 2008

Lab Note 5: The Central Role in Physics, of the Dirac Anticommutator g^uv=(1/2){gamma^u,gamma^v}

Filed under: General Relativity,Gravitation,Physics,Science — Jay R. Yablon @ 11:38 pm

   I would like to take a break from my current work on Kaluza-Klein, and focus on the central importance to physics of the Dirac anticommutator relationship \eta ^{\mu \nu } \equiv {\tfrac{1}{2}} \left(\gamma ^{\mu } \gamma ^{\nu } +\gamma ^{\nu } \gamma ^{\mu } \right)\equiv {\tfrac{1}{2}} \left\{\gamma ^{\mu } ,\gamma ^{\nu } \right\}, when generalized to a non-zero gravitational field in the form g^{\mu \nu } \equiv{\tfrac{1}{2}} \left\{\Gamma ^{\mu } \Gamma ^{\nu } +\Gamma ^{\nu } \Gamma ^{\mu } \right\}\equiv {\tfrac{1}{2}} \left\{\Gamma ^{\mu } ,\Gamma ^{\nu } \right\}.  In particular, when g^{\mu \nu } \ne \eta ^{\mu \nu } but rather include a gravitational field g^{\mu \nu } (x)=\eta ^{\mu \nu } +\kappa h^{\mu \nu } (x), then also the \Gamma ^{\mu } \ne \gamma ^{\mu } , but rather include a “square root” gravitational field h^{\mu } (x) which may be defined as \Gamma ^{\mu } (x)\equiv \gamma ^{\mu } +\kappa h^{\mu } (x).  Combining all the foregoing, this means that \kappa h^{\mu \nu } \equiv {\tfrac{1}{2}} \kappa \left[h^{\mu } \gamma ^{\nu } +\gamma ^{\mu } h^{\nu } +h^{\nu } \gamma ^{\mu } +\gamma ^{\nu } h^{\mu } \right]+{\tfrac{1}{2}} \kappa ^{2} \left[h^{\mu } h^{\nu } +h^{\nu } h^{\mu } \right].

   We also note that in perturbation theory, non-divergent perturbative effects are, in the end, captured in a correction to the vertex factor given by \overline{{\rm u}}(p)\gamma ^{\mu } {\rm u(p)}\to \overline{{\rm u}}(p)\left(\gamma ^{\mu } +\Lambda ^{\mu } \right){\rm u(p)} operating on a Dirac spinor {\rm u(p)}.  That is, the bare vertex \gamma ^{\mu } becomes the dressed vertex \gamma ^{\mu } +\Lambda ^{\mu } .  By then associating the perturbative \Lambda ^{\mu } with \kappa h^{\mu } just specified,  we raise the possibility that gravitational and perturbative descriptions of nature may in some way be interchangeable.  More to the point: when we consider perturbative effects in particle physics, we may well be considering gravitational effects without knowing that this is what we are doing.  The inestimable benefit of gravitational theory over  perturbation theory is that it is non-linear and exact.  The inestimable benefit of perturbation theory over gravitational theory is that we know something about how to achieve its renormalization.  Perhaps by developing this link further via the vitally-central physical relationship g^{\mu \nu } \equiv {\tfrac{1}{2}} \left\{\Gamma ^{\mu } ,\Gamma ^{\nu } \right\}, we can infuse the exact, non-linear character of gravitational theory into perturbation theory, and the renormalizability of perturbation theory into gravitational theory.  Recognizing that “Lab Notes” is in the nature of a scientific diary, this, in any event, is the starting point for this lab note.

   Now, there are two main directions in which to exploit the connection g^{\mu \nu } \equiv {\tfrac{1}{2}} \left\{\Gamma ^{\mu } ,\Gamma ^{\nu } \right\}, and both need to be considered.  First, we may start with the metric tensor g_{\mu \nu } from a known, exact solution to Einstein’s equation, calculate its associated \Gamma ^{\mu } , and then employ \Gamma ^{\mu } in the Dirac equation, in the form 0=\left(\Gamma ^{\mu } \left(i\partial _{\mu } +eA_{\mu } \right)-m\right)\psi .  Using the Schwarzschild solution as the basis, I have done this in detail, in a paper linked at Magnetic Moment Anomalies of the Charged Leptons.  If you would like an “Executive Summary” of this paper, you may obtain this at What the Magnetic Moment Anomaly May Tell Us About Planck-Scale Physics.  What is especially noteworthy, is that the magnetic moment anomaly can perhaps be understood as a symptom of gravitational effects near the Planck scale.

   I actually wrote the above detailed paper in September, 2006, but never posted it anywhere, because as soon as it was written, I went off into writing the related ArXiV paper at titled Ward-Takahashi Identities, Magnetic Anomalies, and the Anticommutation Properties of the Fermion-Boson Vertex.  This paper illustrates the second direction in which to exploit the connection g^{\mu \nu } \equiv {\tfrac{1}{2}} \left\{\Gamma ^{\mu } ,\Gamma ^{\nu } \right\}.  Here, we start with a known \Gamma ^{\mu } , then use g^{\mu \nu } \equiv {\tfrac{1}{2}} \left\{\Gamma ^{\mu } ,\Gamma ^{\nu } \right\}  to obtain the associated g^{\mu \nu } , and then use this as a metric tensor in the usual way.  In this paper, in particular, we start with the perturbative vertex factor \Gamma ^{\mu } \equiv \gamma ^{\mu } +\Lambda ^{\mu } =F_{1} \gamma ^{\mu } +{\tfrac{1}{2}} F_{2} i\sigma ^{\mu \nu } (p'-p)_{\nu } from equation (11.3.29) of Weinberg’s definitive treatise The Quantum Theory of Fields, then we calculate the anticommutators g^{\mu \nu } \equiv {\tfrac{1}{2}} \left\{\Gamma ^{\mu } ,\Gamma ^{\nu } \right\}, and then we use the g^{\mu \nu } as a metric tensor.  What is fascinating about this approach, is that the \Gamma ^{\mu } (p_{\mu } ) are specified in momentum space, rather than spacetime.  This means that g^{\mu \nu } =g^{\mu \nu } (p_{\mu } ) deduced therefrom define a non-Euclidean momentum space, rather than a non-Euclidean spacetime.  This may open up a whole new branch of physics dealing with — I’ll say it again — Non-Euclidean Momentum Space.  As we know from Heisenberg, spacetime is conjugate to momentum space, \left[x^{\mu } ,p^{\nu } \right]=i\hbar \delta ^{i\mu \nu } .  The central result of this paper, arrived at via the Ward-Takahashi Identities which are central to renormalization, is that interaction vertexes are a measure of curvature in momentum space, and the strength of the interaction at a vertex is proportional to the momentum space curvature, see Figures 1 and 2.  This may place particle physics onto a firm geometric footing, but rooted in the geometry of momentum space.

   What I have not yet gotten to, is the question of how to use \left[x^{\mu } ,p^{\nu } \right]=i\hbar \delta ^{i\mu \nu } , as between a non-Eclidean spacetime and a non-Euclidean momentum space,  because in the real world, we have both.  That is a project for the remaining free time in my day, between 3AM and 5AM.  😉

   As always, these are lab notes, representing “work in progress.”  I welcome comments and contributions, as always.


Thesis Defense of the Kaluza-Klein, Intrinsic Spin Hypothesis — EARLY DRAFT

I have been engaged in a number of Usenet and private discussions about the paper Intrinsic Spin and the Kaluza-Klein Fifth Dimension, Rev 3.0 which I posted here on this blog on March 30.

A number of critiques have been raised, which you can see if you check out the recent Usenet threads I started related to intrinsic spin under the heading “Query about intrinsic verus [sic] orbital angular momentum,” over at sci.physics.foundations, sci.physics.relativity and sci.physics.research. These are among the “links of interest” provided in the right-hand pane of this weblog.

I believe that these critiques can be overcome, and that this hypothesis relating to Kaluza-Klein and intrinsic spin and the spatial isotropy of the square of the spin will survive and be demonstrated, ultimately, to be in accord with the physical reality of nature.

I have begun a new paper which is linked at: Thesis Defense of the Kaluza-Klein, Intrinsic Spin Hypothesis, Rev 1.0, which will respond thoroughly and systematically to the various critiques.  What is here so far is the introductory groundwork.  But, I would appreciate continued feedback as this development continues.

Note that the links within the PDF file unfortunately do not work, so to get the intrinsic spin paper, you need to go to Intrinsic Spin and the Kaluza-Klein Fifth Dimension, Rev 3.0.  Also, to get Wheeler’s paper which is referenced, go to Wheeler Geometrodynamics.

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