Sources I am reacting to:
- https://a.rivero.nom.es/claude-on-hans/
- https://www.qeios.com/read/Q7SKTZ
- https://a.rivero.nom.es/research/Areal.pdf
There is a small trick in the de Vries construction which is easy to miss if one jumps too quickly to the electroweak angle. The trick is not, at first, about the weak interaction. It is about a circular relativistic orbit, a Compton clock, and the old quantum habit of replacing a naive angular momentum number by the square root of a Casimir.
The condition is beautifully compact:
\( L_{\rm orb}=\gamma m v r=\hbar\sqrt{s(s+1)}. \)Then comes the odd extra rule. The orbit is asked to close in one Compton period of the orbiting particle:
\( T_{\rm orbit}=T_C(m)={h\over mc^2}. \)For a circular orbit,
\( T_{\rm orbit}={2\pi r\over v}, \)so this says
\( r=\beta {\hbar\over mc},\qquad \beta={v\over c}. \)Put this into the relativistic angular momentum:
\( L_{\rm orb} =\gamma m(\beta c)\left(\beta {\hbar\over mc}\right) =\gamma\beta^2\hbar. \)Therefore the de Vries-Hans condition is simply
\( \gamma\beta^2=\sqrt{s(s+1)}, \)or, spelling out the Lorentz factor,
\( {\beta^2\over\sqrt{1-\beta^2}}=\sqrt{s(s+1)}. \)This is the whole machine. It is tiny. It is also strangely effective.
Squaring gives
\( {\beta^4\over 1-\beta^2}=s(s+1). \)If we set
\( C=s(s+1), \)then the solution for \(\beta^2\) is
\( \beta^2=X©, \qquad X©={\sqrt{C^2+4C}-C\over 2}. \)Now use the two angular momenta which look natural for the weak boson story:
\( s={1\over 2},\qquad s=1. \)Their Casimirs are
\( C_{1/2}={3\over 4}, \qquad C_1=2. \)So
\( \beta_{1/2}^2=X\left({3\over 4}\right)={\sqrt{57}-3\over 8}, \)and
\( \beta_1^2=X(2)=\sqrt{3}-1. \)Then comes the famous de Vries number:
\( \sin^2\theta_{dV} =1-{\beta_{1/2}^2\over\beta_1^2} =0.2231013223\ldots \)This is close to the on-shell weak mixing angle,
\( \sin^2\theta_W^{\rm OS}=1-{M_W^2\over M_Z^2}. \)The magic, if it is magic, is not hidden in a long calculation. It is right there in the sentence: use a relativistic circular orbit, close it in one Compton period, and quantize the angular momentum by \(\sqrt{s(s+1)}\).
The funny part: whose Compton scale is it?
The Qeios note on areal velocity is similar enough to make one suspicious. But it puts the Compton length on the other mass.
In the de Vries-Hans construction, the Compton period belongs to the orbiting particle. If the orbiting mass is \(m\), the special length is
\( L_m={\hbar\over mc}. \)The orbit radius is
\( r=\beta L_m. \)In the areal-speed note, the special length belongs to the source, the center of force. If the source mass is \(M\), the special length is
\( L_M={\hbar\over Mc}. \)The central variable there is the areal velocity,
\( \dot A={J\over 2m}. \)For an ordinary circular Newtonian gravitational orbit,
\( \dot A={rv\over 2}={1\over 2}\sqrt{GMr}. \)Solving for \(r\) gives
\( r={4\dot A^2\over GM}. \)Now introduce the Planck areal speed
\( \dot A_P={L_P^2\over T_P}=cL_P={\hbar\over M_P}. \)Put
\( \dot A=k\dot A_P. \)Then
\( r=4k^2{\hbar\over Mc}=4k^2L_M. \)So if
\( \dot A={1\over 2}\dot A_P, \)then
\( r=L_M. \)This is a very neat cancellation. Newton’s constant disappears. The orbit sits at the reduced Compton length of the source.
So the two constructions can be placed side by side:
\( \hbox{de Vries-Hans:}\qquad r=\beta L_m. \) \( \hbox{Areal-speed note:}\qquad r=L_M. \)The same-looking move is being made, but the Compton length has changed ownership.
In one story, the clock is carried by the orbiting particle. In the other, the scale is set by the center of force.
This is the little crack where a stringy thought can enter.
A de Vries string?
String theory naturally couples to gravity. So if one wants to stringify the de Vries idea, the areal-speed version may be the more natural doorway. Gravity likes the center of force. De Vries likes the orbiting particle. A string can connect the two.
Imagine a string stretched between a source-like object of mass \(M\) and an orbiting endpoint or defect of mass \(m\). The de Vries side wants
\( r=\beta L_m. \)The gravitational areal-speed side wants
\( r=L_M. \)If both descriptions are imposed on the same radius, then
\( \beta L_m=L_M. \)That gives
\( \beta {\hbar\over mc}={\hbar\over Mc}, \)so
\( \beta={m\over M}. \)This is almost too cute. The orbital speed becomes a mass ratio.
But string theory has another standard move: a radius can be exchanged with an inverse radius. In the simplest schematic form, T-duality says something like
\( R\longleftrightarrow {\alpha’\over R}. \)So a more stringy matching would be
\( \beta L_m={\alpha’\over L_M}. \)Then
\( \alpha’=\beta L_mL_M. \)Writing \(\ell_s^2=\alpha’\), this becomes
\( \ell_s^2=\beta {\hbar\over mc}{\hbar\over Mc}. \)This is the kind of formula one wants from a de Vries string: the string length is the geometric mean of two Compton lengths, with the de Vries velocity as the dimensionless weight.
That would make the ultraviolet string scale and the infrared electroweak mass scale talk through a duality, rather than through ordinary running alone.
And this is important, because the weak angle produced by de Vries is not naturally a string-scale angle. It is close to the on-shell electroweak angle, the one built from pole masses:
\( \sin^2\theta_W^{\rm OS}=1-{M_W^2\over M_Z^2}. \)String theory, by contrast, normally gives relations among gauge couplings at a very high scale. In old grand-unified language, the famous high-scale number is
\( \sin^2\theta_W={3\over 8}. \)The de Vries value is down near
\( 0.223. \)So a de Vries string needs more than a pretty classical orbit. It needs a UV/IR mechanism.
That mechanism could be one of three things:
- ordinary renormalization-group flow plus threshold matching;
- a T-dual map between source and orbiting Compton lengths;
- a holographic or warped construction where a UV string datum appears as an IR mass relation.
The second option is the strangest and therefore the most fun.
The Casimir clue
Actually, this is the point I should have stressed earlier. The de Vries formula is not really built out of an arbitrary spin label. It is built out of the quadratic angular-momentum operator.
Quantum mechanically,
\( J^2=\hbar^2s(s+1). \)So the de Vries-Hans orbit is better written as
\( L_{\rm orb}=\sqrt{J^2}=\hbar\sqrt{s(s+1)}. \)That matters, because \(J^2\) is exactly the sort of quantity string theory naturally cares about. String states are organized into representations of spacetime rotation or Lorentz symmetry, and those representations are labeled by Casimirs. At the same time, classical rotating strings produce Regge trajectories, schematically
\( J\sim \alpha’ M^2, \)while the quantum spectrum is sorted by oscillator levels and spin representations. So angular momentum squared, representation Casimirs, and mass squared are all part of the native string vocabulary.
This makes the de Vries trick look less like a random classical orbit and more like a translation rule:
\( \hbox{quadratic Casimir}\quad J^2 \quad\longrightarrow\quad \hbox{relativistic orbit magnitude}\quad \sqrt{J^2}. \)The de Vries formula depends on
\( s(s+1). \)This is the \(SU(2)\) quadratic Casimir. For \(s=1/2\) it is \(3/4\). For \(s=1\) it is \(2\).
String worldsheet conformal field theory also loves Casimirs. In an \(SU(2)_k\) Wess-Zumino-Witten model, the conformal weight of a spin-\(j\) primary is
\( h_j={j(j+1)\over k+2}. \)So the de Vries recipe resembles a current-algebra recipe, except that it feeds the Casimir into the nonlinear function
\( X©={\sqrt{C^2+4C}-C\over 2}. \)A naive stringy deformation would replace
\( C\longrightarrow {C\over k+2}. \)Then define
\( \sin^2\theta(k) =1- {X\left({3/4\over k+2}\right) \over X\left({2\over k+2}\right)}. \)The original de Vries value corresponds to
\( k+2=1, \)or
\( k=-1. \)That is provocative. Compact unitary \(SU(2)_k\) models want nonnegative integer \(k\). So \(k=-1\) is a red flag. But red flags are sometimes useful. They tell us where the idea must either become deeper or die.
A viable de Vries string would need an effective sector where the Casimir appears with denominator one. That might be a ghost contribution, a coset remnant, a noncompact cousin, a supergroup trick, or some constrained zero-mode sector. It may also be impossible. Either outcome would teach us something.
What should be calculated first?
Here is the research program I would actually run.
First, forget strings for a week and classify the Casimir transform.
Take
\( \Theta(C_F,C_A)=1-{X(C_F)\over X(C_A)}. \)Now feed it natural pairs of Casimirs from simple groups. Ask whether \(SU(2)\) is special. If many groups and representations give numbers near \(0.223\), the de Vries value becomes less impressive. If the \(SU(2)\) fundamental-adjoint pair is isolated, the result becomes more interesting.
Second, check the current-algebra lift.
Study
\( \Theta_k=1- {X\left({C_F\over k+h^\vee}\right) \over X\left({C_A\over k+h^\vee}\right)}. \)For \(SU(2)\), \(h^\vee=2\). The de Vries value wants \(k=-1\). One should search for known worldsheet sectors whose effective level or effective Sugawara denominator behaves this way.
Third, build the simplest rotating string with a massive endpoint.
The endpoint contribution is
\( J_{\rm end}=\gamma mvr. \)The string itself contributes angular momentum:
\( J_{\rm total}=J_{\rm end}+J_{\rm string}. \)The de Vries condition wants
\( J_{\rm total}\approx\hbar\sqrt{s(s+1)}. \)If \(J_{\rm string}\) is large and unavoidable, the point-particle de Vries orbit is destroyed. If \(J_{\rm string}\) can vanish, cancel, or become topological in a special sector, the idea survives.
Fourth, do the electroweak matching honestly.
A string model gives high-scale gauge data. The measured on-shell angle is a pole-mass quantity. Between them lies the full chain
\( \hbox{string scale} \longrightarrow \hbox{thresholds} \longrightarrow \overline{\rm MS}\hbox{ couplings} \longrightarrow \hbox{pole masses} \longrightarrow \sin^2\theta_W^{\rm OS}. \)A de Vries string only becomes serious if its pretty orbit survives this chain.
A possible punchline
The conservative interpretation is this:
The de Vries number is a remarkable \(SU(2)\) Casimir coincidence dressed up as a relativistic Compton orbit.
The bolder interpretation is this:
The orbit is a shadow of a source-orbit duality. The orbiting-particle Compton scale and the source-particle Compton scale are two descriptions of the same circular zero mode. A string stretched between them supplies the missing duality.
In symbols, the dream is
\( r=\beta L_m \quad\hbox{and}\quad r\sim {\alpha’\over L_M}. \)Then
\( \ell_s^2\sim \beta L_mL_M. \)The de Vries velocity would not merely be a velocity. It would be the dimensionless factor relating two Compton descriptions of one stringy orbit.
This is a strange idea. But it is strange in a useful way. It gives concrete calculations:
- classify the Casimir transform;
- test the effective \(k=-1\) clue;
- compute the rotating-string endpoint correction;
- run the weak angle from the string scale to the pole masses;
- check whether the source-Compton and orbiting-Compton conditions can be made dual rather than merely similar.
The smallest honest slogan is:
\( \boxed{\hbox{de Vries quantizes a Compton orbit; a de Vries string would dualize whose Compton orbit it is.}} \)
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