→ Discover2011:根本沒有磁單極 12/08 23:07
給你這篇算好讀的介紹 這篇我很久以前有讀過,Polchinski是我的偶像,書超難讀><
Steven Weinberg曾說過Joseph Polchinski是他見過最聰明的人,所以我相信Polchinski
後面他含蓄地說
But as I have discussed, the existence of magnetic monopoles seems like one
of the safest bets that one can make about physics not yet seen.
Monopoles, Duality, and String Theory
Authors: Joseph Polchinski
http://arxiv.org/abs/hep-th/0304042
2. The Necessity of Monopoles
One of Dirac’s remarkable discoveries was the connection between magnetic
monopoles and charge quantization.1,4 Very early in the history of quantum
theory, he recognized the important connection between geometry and quantum
mechanics. Dirac showed that in the presence of a magnetic charge g, in order
for the quantum mechanics of an electric charge e to be consistent one had to
have
eg = 2πn.
Thus the existence of even a single magnetic charge forces every electric
charge to be a multiple of 2π/g. From the highly precise electric charge
quantization that is seen in nature, it is then tempting to infer that magnetic
monopoles exist, and indeed Dirac did so:1
One would be surprised if nature had made no use of it.I would like to discuss this from the point of view of the modern search
for a unified theory, and to offer two general principles of completeness:
(1) In any theoretical framework that requires charge to be quantized,
there will exist magnetic monopoles.
(2) In any fully unified theory, for every gauge field there will exist electric
and magnetic sources with the minimum relative Dirac quantum
n = 1
(more precisely, the lattice of electric and magnetic charges is maximal).
Obviously neither of these is a theorem. Rather, they are aesthetic principles
based on experience with a rather wide range of examples. I will give
three examples of the first principle, and two of the second.
2.1. Grand Unification
The most well-known example of the first principle of completeness is the
t'Hooft-Polyakov monopole. If the U(1) of electromagnetism is embedded in a
semisimple group, for example in grand unification
SU(3) ╳咝SU(2) ╳咝U(1) ﹝垡 SU(5) ,
then electric charge is necessarily quantized, since it descends from the
quantized representations of the unified group.
Under precisely these conditions, 删t'Hooft and Polyakov showed that
magnetic monopoles will exist as smooth but topologically nontrivial
classical solutions.
Let me give a brief description of this idea. Dirac showed that the
vector potential for a magnetic charge had a singularity along a string
extending from the charge. Figure 1 shows a Dirac string.
Let us parallel transport a P' P
Figure 1. (自己看paper)
A Dirac string extending from the monopole. charged field on the infinitesimal
→ →
path P around it according to dψ = ie A ．dx ψ讨 .
Because of the string singularity, the field picks up a net phase eg in the
process; this is unobservable precisely if eg = 2n.
One can think of the phase of 讨 as looping around the U(1) group, which is a
circle, n times.Now, if we pull the loop off of the string to the position Ｐ',
the field is nonsingular and the phase is constant. Hence the net phase must
drop rapidly from 2n to zero as the loop is pulled past the monopole, and this
observable phase signifies a singularity in the field. When U(1) is embedded
in a semisimple group, however, the loop can become topologically trivial:
in SU(2) for example, a rotation through 4π纸can be smoothly deformed to
a trivial path. Thus there are smooth field configurations, which at long
distance look like Dirac monopoles, with a net U(1) magnetic charge.
As a postscript to the talk, I should note that this argument does not
require that one can obtain eg = 2纸, and indeed this depends on the field
content. In the Georgi-Glashow model U(1) ﹝垡 SU(2), if there are only
fields of integer isospin then the minimum quantum is eg = 4纸, while if there
are fields of half-integer isospin then eg = 2纸 is obtained.
(This is related to the fact that some simple groups do have a finite set of
nontrivial closed loops.) So in the sense of the second principle I would have
to say that this theory is not fully unified, precisely because its matter
content is not fixed but subject to arbitrary choice.
2.2. Kaluza-Klein Theory
If spacetime is five-dimensional, with the added dimension x4 being
periodic,then five-dimensional gravity gives rise to both gravity and a Maxwell
field in four dimensions. This was perhaps the first application of spontaneous
symmetry breaking as a unifying concept — the laws of physics are invariant
under Lorentz transformations in all five spacetime dimensions,while the state
we live in is invariant only under the four-dimensional symmetry group.
The metric components gμ4 become the Maxwell potential,and gauge invariance
arises from reparameterizations of x4. What the fourdimensional physicist sees
as electric charge is therefore momentum in the 4-direction,and it is quantized
because of the periodicity in this direction.
A Dirac monopole configuration would again have a string, which is now
a coordinate singularity as in Figure 1. An infinitesimal loop in the
fivedimensional geometry, whose projection to four dimensions is the loop P,
makes one or more circuits of the x4 coordinate. An infinitesimal loop away
from the string, such as P′, does not loop the x4 direction. Thus there is
again singular behavior as the loop is pulled past the monopole. However,
Gross, Perry, and Sorkin showed that again there are smooth geometries that
look like Dirac monopoles outside of some core region. The point is that if the
radius of the x4 direction shrinks to zero in an appropriate way,then at the
origin there is only a coordinate singularity and the loop can be smoothly
slid off the string.
接下來2.3. U(1) Lattice Gauge Theory和一些弦論先跳過><
3. Conclusions
3.1. The Existence of Monopoles
By the end of his career Dirac became less certain about the existence of
monopoles. He forgot his earlier dictum to ignore experiment!
But as I have
discussed, the existence of magnetic monopoles seems like one of the safest
bets that one can make about physics not yet seen. It is very hard to predict
when and if monopoles will be discovered. If their mass is at the grand unified
scale as one expects, then they will be beyond the reach of accelerators,
while inflation has almost certainly diluted any primordial monopoles beyond
discovery. It is curious to contemplate this unfortunate situation, where
theory predicts the existence of an object
(and its production,but in experiments that can only be carried out in thought)
and at the same time suggests that it may never be seen. But we must continue
to hope that we will be lucky, or unexpectedly clever, some day.
3.2. Duality and Beauty
Many of the electric and magnetic objects that I have discussed look quite
different from one another, but there is strong evidence that in each case
that there are dualities that interchange them. For Dirac this would have
been a triviality. He notes that his theory of pointlike electric and magnetic
charges is invariant under the interchange of the two objects, along with
the interchange of electric and magnetic potentials. Dirac’s theory is rather
formal, since the magnetic coupling is the inverse fine structure constant,
but one can regard the lattice as providing a precise definition of a cutoff
theory, and with appropriate choice of the action it is self-dual.
For the grand unified and Kaluza-Klein cases, however, any duality
must be quite nontrivial. In these cases the electric charges are pointlike
quanta, while the magnetic charges are smooth classical configurations. To be
precise, this is the picture at weak coupling. Now it would not be surprising
that as the coupling is turned up the electric objects begin to emit pairs and
become big and fuzzy like the solitons. The great surprise (duality) is that
when the coupling becomes very large the magnetic objects become more and more
pointlike and the theory can be described in terms of their local fields.
It is a remarkable property of the quantum theory that the degrees of
freedom can, at least with some assistance from supersymmetry,reorganize
themselves in this way. Indeed, we do not fully understand the details of this,
but the number of independent consistency checks isenormous.
Even further, in string theory all of the electric and magnetic objects
that have been discussed here, with the (possible) exception of the lattice
examples, are related to one another by dualities. These examples involve
widely different aspects of gauge field geometry, spacetime geometry,
‘stringy’ geometry, string perturbation theory, and quantum and classical
physics. The existence of a single structure that unifies such a broad range
of physical and mathematical ideas, and many others as well, is unexpected
and remarkable. Earlier I declined to define beauty, but one can recognize it
when one sees it, and here it is. This is one illustration of why the scientific
path that Dirac laid out has been such a fruitful one in recent times.
→ baal678:那有重力藍移嗎 12/09 00:22
有,但是不要聽Discovery2011胡說八道
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