charge quantization and gravity
Martin Lohmann
When Dirac showed that the existence magnetic monopole would somehow
through nontrivial homotopy groups imply the quantization of electric
charge, many physicists believed that this possibility would be that
one that is realized in nature because there were no other explanation
for it and because a topological explanation sounded somewhat
"beautiful".
Now, electric charge is, in the Weinberg model of electroweak
interactions and in the standard model, linked to weak isospin and
weak hypercharge. If you fix this quantities for particles, then
electric charge is naturally quantized. It now happens that there are
several constraints on the values of weak isospin and weak hypercharge
due to anomaly cancellation which, in the standard model without
gravity, fixes these numbers up to a constant.
But if you also include the simplest model of gravity in the standard
model, you get another constraint due to gravitational anomalies which
totally fixes the values of isospin and hypercharge. So this extension
of the standard model leads naturally to a charge quantization, just
as the magnetic monopole.
My question is now, why do physicists not believe in this "extended"
standard model, while they did believe in the magnetic monopole? Is it
just because of renormalization?
Hendrik van Hees
Martin Lohmann wrote:
> Now, electric charge is, in the Weinberg model of electroweak
> interactions and in the standard model, linked to weak isospin and
> weak hypercharge. If you fix this quantities for particles, then
> electric charge is naturally quantized. It now happens that there are
> several constraints on the values of weak isospin and weak hypercharge
> due to anomaly cancellation which, in the standard model without
> gravity, fixes these numbers up to a constant.
I always wondered, if there is only this one pattern (of course up to a
constant) of charges or whether there exist others. Do you have
references, where these questions are discussed?
--
Hendrik van Hees Cyclotron Institute
Phone: +1 979/845-1411 Texas A&M University
Fax: +1 979/845-1899 Cyclotron Institute, MS-3366
http://theory.gsi.de/~vanhees/ College Station, TX 77843-3366
Arvind Rajaraman
martin_...@t-online.de (Martin Lohmann) wrote in message news:<f38a1178.04041...@posting.google.com>...
> Now, electric charge is, in the Weinberg model of electroweak
> interactions and in the standard model, linked to weak isospin and
> weak hypercharge. If you fix this quantities for particles, then
> electric charge is naturally quantized. It now happens that there are
> several constraints on the values of weak isospin and weak hypercharge
> due to anomaly cancellation which, in the standard model without
> gravity, fixes these numbers up to a constant.
> But if you also include the simplest model of gravity in the standard
> model, you get another constraint due to gravitational anomalies which
> totally fixes the values of isospin and hypercharge.
This is not correct. In four dimensions, the gauge and gravitational
anomalies come from triangle diagrams which involve either one or
three chiral currents. The gauge anomalies involve the U(1)
hypercharge and the SU(2) gauge bosons. The nontrivial anomalies are
the ones with U(1)-SU(2)-SU(2), U(1)-U(1)-U(1), and
SU(2)-SU(2)-SU(2). The last vanishes always. The first two vanish if
the sum of hypercharges in a generation is zero, and the sum of the
cubes is also zero. These happen to be satisfied.
The only gravitational anomaly is U(1)-graviton-graviton, which tells
us again that the sum of hypercharges in a generation is zero. So we
get no new constraints.
Furthermore, it is clear that multiplying all the hypercharges by a
constant cannot affect the vanishing of the anomalies. So we cannot
get charge quantization from anomaly considerations alone.
Igor
martin_...@t-online.de (Martin Lohmann) wrote in message news:<f38a1178.04041...@posting.google.com>...
Which "extended" standard model are you referring to? There's String
Theory, Loop Quantum Gravity, M-Theory, or Supergravity, just to name
a few of the competing models that are out there these days. And just
as a point, not everyone buys into the existence of magnetic
monopoles, either. In fact, I think those that do are very much in
the minority. The bottom line is that all of these ideas are still
severely lacking in any real experimental/observational support.
Arvind Rajaraman
> martin_...@t-online.de (Martin Lohmann) wrote in message news:<f38a1178.04041...@posting.google.com>...
> > Now, electric charge is, in the Weinberg model of electroweak
> > interactions and in the standard model, linked to weak isospin and
> > weak hypercharge. If you fix this quantities for particles, then
> > electric charge is naturally quantized. It now happens that there are
> > several constraints on the values of weak isospin and weak hypercharge
> > due to anomaly cancellation which, in the standard model without
> > gravity, fixes these numbers up to a constant.
> > But if you also include the simplest model of gravity in the standard
> > model, you get another constraint due to gravitational anomalies which
> > totally fixes the values of isospin and hypercharge.
>
> This is not correct. In four dimensions, the gauge and gravitational
> anomalies come from triangle diagrams which involve either one or
> three chiral currents. The gauge anomalies involve the U(1)
> hypercharge and the SU(2) gauge bosons. The nontrivial anomalies are
> the ones with U(1)-SU(2)-SU(2), U(1)-U(1)-U(1), and
> SU(2)-SU(2)-SU(2). The last vanishes always. The first two vanish if
> the sum of hypercharges in a generation is zero, and the sum of the
> cubes is also zero. These happen to be satisfied.
I should make the above statements precise. Actually, the first
vanishes if the sum of hypercharges of the left handed particles is
zero, and the second if the sum of the cubes of the left handed
particles minus that of the right handed particles is zero. So the
gravitational anomaly is actually new, it says that the sum of all the
hypercharges in a generation is zero.
> Furthermore, it is clear that multiplying all the hypercharges by a
> constant cannot affect the vanishing of the anomalies. So we cannot
> get charge quantization from anomaly considerations alone.
This statement remains valid.
Alfred Einstead
> Now, electric charge is, in the Weinberg model of electroweak
> interactions and in the standard model, linked to weak isospin and
> weak hypercharge. If you fix this quantities for particles, then
> electric charge is naturally quantized. It now happens that there are
> several constraints on the values of weak isospin and weak hypercharge
> due to anomaly cancellation which, in the standard model without
> gravity, fixes these numbers up to a constant.
The "several constraints" do NOT fix the charge, not even
assuming generation-independence. They only fix an undetermined
linear combination of the quantum numbers
G = (Baryon-Lepton)/2, and Y = Weak Isospin.
In order to recover Y, you also need to have the additional
assumption that:
* there exists at least one sterile fermion sector.
These would be the right-neutrinos and left-antineutrinos.
More generally, you have both Y and G and (as alluded to elsewhere)
there are other regularities that supersede the considerations
you're rasing and point in an entirely different direction. In
particular: (1) the 5 quantum numbers a, b, c, d, e:
For U(2):
a, b: Y/g' - G +/- I3/g
For U(3):
c: G + sqrt(4/3) L3/gs
d, e: G - sqrt(1/3) L3/gs +/- L8/gs
all assume only the values +/- 1/2 over the fermion spectrum;
with all 32 combinations giving you each of the particles of
each generation; (2) all flavor-changing interactions are
of the form
For U(2):
a up b down <-> a down b up
For U(3):
c up d down <-> c down d up
d up e down <-> d down e up
e up c down <-> e down c up
(3) the 2 invariants are those associated with the U(2) x U(3)
structure alluded to above:
For U(2):
3 (Y/g' - G)^2 + (I1^2 + I2^2 + I3^2)/g^2 = 3/4
For U(3):
6 G^2 + (L1^2 + ... + L8^2)/gs^2 = 3/2
which means it's actually Y' = Y/g' - G, I3, G, L3, L8 that are
the natural quantities to consider, rather than Y, I3, L3 and L8.
In particular (4) they are mutually orthogonal over the fermion
spectrum:
tr(Y' I3) = tr(Y' G) = tr(Y' L3) = tr(Y' L8) = 0
tr(I3 G) = tr(I3 L3) = tr(I3 L8) = 0
tr(G L3) = tr(G L8) = tr(L3 L8) = 0.
There's a deeper level of quantization at work here -- the 5-fold
q-bit quantization described above which bespeaks an underlying
U(2) x U(3) symmetry group -- all of which supersedes the Dirac
argument.
Martin Lohmann
> constant) of charges or whether there exist others. Do you have
> references, where these questions are discussed?
Well, I learned about constraints on these values due to anomaly
cancellation in "Effective Lagrangians for the Standard Model" by A.
Dobado et al., but this wont help you too much, because it treats only
this pattern of cancellation of gravitational anomalies and usual
standard mdel anomalies.
But I cant really think of another way to limit the number possible
choices of these constants, because they are nothing but coupling
constants of the SU(2)_L x U(1) electroweak gauge group and are
therefore classically not constrained, but on a quantum level only due
to anomalies (which is, as far as I know, the only striking difference
between classical and quantum symmetries).
Martin Lohmann
> anomalies come from triangle diagrams which involve either one or
> three chiral currents. The gauge anomalies involve the U(1)
> hypercharge and the SU(2) gauge bosons. The nontrivial anomalies are
> the ones with U(1)-SU(2)-SU(2), U(1)-U(1)-U(1), and
> SU(2)-SU(2)-SU(2). The last vanishes always. The first two vanish if
> the sum of hypercharges in a generation is zero, and the sum of the
> cubes is also zero. These happen to be satisfied.
>
> The only gravitational anomaly is U(1)-graviton-graviton, which tells
> us again that the sum of hypercharges in a generation is zero. So we
> get no new constraints.
>
> Furthermore, it is clear that multiplying all the hypercharges by a
> constant cannot affect the vanishing of the anomalies. So we cannot
> get charge quantization from anomaly considerations alone.
As far as I know, there are the anomalies due to SU(3), SU(2)_L and
U(1) gauge symmetry and due to gravitational effects (to be more
precise, the gravitational anomalies are present if you take the
ordinary standard model lagrangian and make it covariant under general
spacetime diffeomorphisms by replacing all derivatives with their
general relativistic counterparts; quantizing this is another
problem). SU(3) anomaly cancellation implies that
0 = SUM y_L - y_R
where the sum goes over all quark families and L and R denote the
left-and right- handed components of hypercharge. From SU(2)_L we get
0 = 3 x SUM_1 y_L + SUM_2 y_L
where SUM_1 goes over quark families and SUM_2 over lepton families.
The U(1) anomaly gives
0 = 3 x SUM_1 y_L^3 - y_R^3 + SUM_2 y_L^3 - y_R^3.
These conditions do determine the hypercharges up to a constant which
can be obtained by setting the electrons electric charge equal to -1.
But there is another anomaly, namely the gravitational anomaly, which
gives another constraint,
0 = 3 x SUM_3 y_L - y_R + SUM_4 y_L - y_R
where SUM_3 is over all quarks and SUM_4 is over all leptons. This
equations fully determines the hypercharges. Of course, it seems
strange to me how one can determine the electrons charge by purely
theoretical methods, but when I read this, it seemed consistent
(please correct me if I am wrong).