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advanced_notions:topological_defects:magnetic_monopoles [2017/12/20 10:56]
jakobadmin created
advanced_notions:topological_defects:magnetic_monopoles [2019/07/11 17:42] (current)
jakobadmin [Why is it interesting?]
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 <tabbox Why is it interesting?> ​ <tabbox Why is it interesting?> ​
  
-If magnetic monopoles exist, they explain why electric charge is quantized. Moreover, magnetic monopoles appear automatically in extensions of the standard model, such as [[theories:speculative_theories:​grand_unified_theories|grand unified theories]]. ​+If magnetic monopoles exist, they explain why electric charge is quantized. Moreover, magnetic monopoles appear automatically in extensions of the standard model, such as [[models:speculative_models:​grand_unified_theories|grand unified theories]]. ​
  
 +
 +----
 +
 +<​blockquote>​I am quite certain that magnetic monopoles really exist. How, when and if they will be found is another matter.
 +<​cite>​[[https://​cds.cern.ch/​record/​246430?​ln=en|Polyakov]]</​cite></​blockquote>​
 <tabbox Layman> ​ <tabbox Layman> ​
  
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 <tabbox Student> ​ <tabbox Student> ​
  
-<​blockquote>​ +There are two types of monopoles. One type is called Dirac monopoles and should be considered as a toy model, because there are many problems with such objects. Nevertheless,​ Dirac monopoles are great to get familiar with many important notions that are needed to describe more complicated topological objects. Dirac monopoles can be added to a pure $U(1)$ Yang-Mills theory, i.e. a theory with only one gauge boson present. Dirac monopoles are not topological solitons, because the electromagnetic field is singular at the location of the monopole and therefore the field energy is infinite. ​
-Ever since Dirac has proposed the Dirac monopole +
-generalizing the Maxwell’s theory, the monopole has become +
-an obsession theoretically and experimentally [1]. +
-After the Dirac monopole we have had the Wu-Yang +
-monopole [2], the ’t Hooft-Polyakov monopole [3], the +
-grand unification (Dokos-Tomaras) monopole [4], and the +
-electroweak (Cho-Maison) monopole [5–9]. But none of +
-them except for the electroweak monopole might become +
-realistic enough to be discovered. +
- +
-<​cite>​https://​arxiv.org/​pdf/​1703.02188.pdf</​cite>​ +
-</​blockquote>​ +
- +
-There are two types of monopoles. One type is called Dirac monopoles and should be considered as a toy model, because there are many problems with such objects. Nevertheless,​ Dirac monopoles are great to get familiar with many important notions that are needed to describe more complicated topological objects. Dirac monopoles can be added to a pure $U(1)$ Yang-Mills theory, i.e. a theory with only one gauge boson present. Dirac monopoles are not topological ​[[quantum_field_theory:​notions_solitons|solitons]], because the electromagnetic field is singular at the location of the monopole and therefore the field energy is infinite. ​+
  
 The other type is called 't Hooft-Polyakov monopole. These are topological solitons and correspond to a field configuration with finite energy. They exist in gauge theories with, for example $SU(2)$ gauge symmetry, where the scalar field breaks the symmetry. ​ The other type is called 't Hooft-Polyakov monopole. These are topological solitons and correspond to a field configuration with finite energy. They exist in gauge theories with, for example $SU(2)$ gauge symmetry, where the scalar field breaks the symmetry. ​
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 -->The Dirac Monopole# -->The Dirac Monopole#
  
 +A Dirac monopole is an object that can be introduced in a pure U(1) Yang-Mills theory, i.e. in a theory where only gauge fields are present. In contrast to the other topological objects that we will discuss it is not a so-called soliton. A soliton is defined as an topological object with finite field energy. A Dirac monopole has infinite field energy, like, for example, an electron. Thus a Dirac monopole is an object like the familiar elementary particles, whereas some solitons can't be interpreted as a particle at all!
  
 +A Dirac monopole does not care about time and thus we are only interested in its description in space $\mathbb{R}^3$. ​ The magnetic point charge introduces a singularity,​ which means that the electromagnetic field becomes infinite at the location of the monopole. Thus, we must restrict our description to $\mathbb{R}^3 - \{0 \} \simeq S^2$. The effect of the monopole is that we can't define our vector potential consistently everywhere on $S^2$. Instead, we must use (at least) two local patches, for example, two hemispheres. In the region where the patches overlap, we must have a function, called a transitions function, that transforms the description of the vector potential in one patch into the description in the other patch. In the most minimalistic case, the overlap is topologically simply a circle on the sphere $S^2$, i.e. $S^1$. Our transition function is therefore a map $g(x)$ that yields for each point of the overlap region, i.e. on the spatial circle $S^1$, the gauge transformation that correctly connects our two patches. Thus, the transition function is a map from $S^1$ to $U(1)$.
 +
 +Each Lie group is also a manifold, and $U(1)$ is simply the set of unit complex numbers and therefore simply the unit circle $S^1$. Therefore, we can say that the transition function is a map from the spatial overlap region $S^1$ to the gauge group $S^1$. Now depending on how many monopoles are present in our system, the transition function changes. This means directly that all information about the monopole or the monopoles is encoded in the transition function, which is a map $S^1 \to S^1$.
 +
 +When no monopole is present in the system, our transition function is simply the trivial map, that maps each point of the spatial circle onto the same point on $S^1$. If there is exactly one monopole present, we find instead that our transition function is a one-to-one map, i.e. each point of the spatial circle is mapped onto exactly one point of the gauge circle $S^1$. When two monopoles are present in our system, our transition function is a two-to-one map. This means that when we move once around the spatial circle $S^1$, we meet every element of the gauge circle $S^1$ twice. In this sense, the number of monopoles is characterized by an integer number, called the winding number.
 +
 +In topology terms, we say that a Dirac monopole is characterized by the first homotopy $\pi_1$ group of $U(1) \simeq S^1$. This homotopy group is $\pi_1(S^1)=\mathbb{Z}$,​ the integers, which is exactly what we use to label monopoles.
 +
 +----
 There are many good reason to not introduce magnetic monopoles in the standard $U(1)$ gauge theory of electromagnetism. However, Dirac did it anyway and realized that we don't need a globally well defined vector potential. Instead it is sufficient if $\vec A$ is a connection. ​ There are many good reason to not introduce magnetic monopoles in the standard $U(1)$ gauge theory of electromagnetism. However, Dirac did it anyway and realized that we don't need a globally well defined vector potential. Instead it is sufficient if $\vec A$ is a connection. ​
  
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   * See also the great discussion in section 3.4 in Topological Solitons by Manton and Sutcliffe and    * See also the great discussion in section 3.4 in Topological Solitons by Manton and Sutcliffe and 
   * http://​physics.stackexchange.com/​a/​167160/​37286   * http://​physics.stackexchange.com/​a/​167160/​37286
 +
 +
 <-- <--
  
 --> ’t Hooft–Polyakov Monopoles # --> ’t Hooft–Polyakov Monopoles #
 +
 +In contrast to the Dirac monopole, the  't Hooft-Polyakov monopole is a topological soliton, i.e. has finite field energy. It appears in $SU(2)$ or $SO(3)$ Yang-Mills theory, where symmetry breaking happens through scalar fields.
 +
 +The 't Hooft-Polyakov monopole also does not care about time and thus we again investigate its behaviour in spatial $\mathbb{R}^3$ only.
 +
 +We assume that spontaneous symmetry breaking happens, which means some scalar field $\phi$ gets a non-zero vacuum expectation value $\langle \phi \rangle \neq 0$.
 +Now, the important thing is that there is, in general, not a unique vacuum expectation value that minimizes the scalar potential. Instead, there is an infinite number of possible vev orientations that also minimize the potential, because we can act on the scalar field with gauge transformations. The Lagrangian is invariant under gauge transformations and thus when some scalar field configuration $\langle \phi \rangle$ minimizes the scalar potential, the gauge transformed field configuration $g\langle \phi \rangle$ also minimizes the potential.
 +
 +Finite energy means for the scalar field only that at spatial infinity, we must have
 +
 +$$ |\langle \phi \rangle |= v^2 . $$
 +
 +This condition is fulfilled with $\langle \phi \rangle = (0,0,v)$, but also, for example, with  $\langle \phi \rangle = (v,0,0)$. These two scalar field configurations are connected by a gauge transformation.
 +
 +The set of all possible vevs is called the vacuum manifold:
 +
 +$$ M \equiv \{ \phi : V(\phi)=0 \}. $$
 +
 +The structure of the vacuum manifold and thus of all possible vevs can be understood as follows.
 +
 +As mentioned above, we can in general transform a given scalar field configuration $\phi_0$ that minimizes the scalar potential with a gauge transformation $g\phi \equiv \phi^g_0$ and get again a scalar field configuration that minimizes the scalar potential. However, not all $\phi^g_0$ are necessarily distinct. In general, there is some subgroup $H$ that does not change $\phi^g_0$ at all. (This subgroup $H$ characterizes the unbroken subgroup.) Explicitly: For some $h \in H$, we have
 +
 +$$ \phi^h_0 ​ =\phi_0.$$
 +
 +An explicit example would be when our scalar field configuration is $(0,0,v)$, all gauge transformation that only affect the first two components, have no effect.
 +
 +To understand the structure of the vacuum manifold, we want to get rid of these redundant field configurations. Mathematically,​ this is done by "​modding out" the gauge transformations that do not change anything at all. This idea comes about as follows:
 +
 +Given an element $g$, we know that it has exactly the same effect as the combination of gauge transformations $g h$, for all $h \in H$, because, well, by definition $H$ is the subgroup of transformations that have no effect on $\phi_0$:
 +
 +$$ g h \phi_0 = g \phi_0 = \phi_0^g \quad \forall h \in H. $$
 +
 +Thus, we get those gauge transformations that have a truly distinct effect by grouping together each $g$ with those gauge transformations that we can get from $g$ by acting on it from the right with some element of $H$. As explained above, we do this because the effect of some $g$ and each other gauge transformation that we can get from $g$ by acting on it with some element of $H$ has exactly the same effect on $\phi_0$ and thus does not yield anything new. We will then treat each of these sets of gauge transformations as a single element of the vacuum manifold.
 +
 +Mathematically,​ this is called the coset space $G/H$:
 +
 +$$ \{gH : g \in G \} \equiv G/H. $$
 +
 +Each element of this coset space is called a coset and is a set of gauge transformations that has exactly the same effect on $\phi_0$. Each coset is a point on our vacuum manifold. Therefore, when a group $G$ breaks to a subgroup $H$ the vacuum manifold $M$ is given by
 +
 +$$ M = G/H  $$
 +
 +Explicitly, for the breaking of $SU(2)$ to $U(1)$, we have $SU(2)/​U(1)=S_{vac}^2$,​ i.e. our vacuum manifold is two-sphere.
 +
 +To get a field configuration with finite energy, the only thing that we require is that the scalar field is in a configuration at spatial infinity that minimizes the scalar potential. Thus a possible field configuration with finite energy is the trivial configuration
 +
 +$$ \phi^a = (0,0,v). $$
 +
 +everywhere at spatial infinity (i.e. on the sphere $S^2$ with radius $r = \infty$).
 +However, equally possible are more complicated and interesting configurations,​ for example:
 +
 +$$ \phi^a \to^{r\to \infty} \frac{v r^a}{|r|}. ​ $$
 +
 +This scalar field configuration looks as follows
 +
 +[{{ :​advanced_notions:​topological_defects:​cs_monopole.gif?​nolink&​400|(a) is the hedgehog monopole configuration and (b) a trivial configuration.}}]
 +
 +and is thus called "the hedgehog"​.
 +
 +Again, the scalar field at spatial infinity takes only values in the vacuum manifold (i.e. field configuration that minimize the scalar potential): $\phi^a \phi^a = v^2$.  When we now talk a walk on the sphere $S^2$ at spatial infinity such that we touch each point on this sphere exactly ones, we notice that on this path the scalar field configurations also takes each value on the possible vacuum manifold $S_{vac}^2$ exactly once, too. In this sense, this field configuration is defined by a the winding number $1$. This field configuration defines a one-to-one map from the spatial sphere at infinity $S^2$ to the vacuum manifold $S_{vac}^2$. In contrast, for the "​trivial"​ field configuration,​ where $ \phi^a = (0,0,v) $ everywhere on $S^2$, all elements of the spatial sphere at infinity get mapped to the same element on $S_{vac}^2$ and thus this map has winding number $0$.
 +
 +The important topological idea is now that there is no global gauge transformation that transforms the scalar field configuration with winding number $1$ into the trivial one with winding number $0$. This is known as the statement that "you can't comb the hedgehog"​. This means that we can't find a gauge transformation that is defined everywhere on $S^2$ and transforms the hedgehog field configuration into the unitary gauge, where the scalar field points in the same direction everywhere on $S^2$. There is a gauge transformation that almost achieves this, however it is singular at the southpole and is therefore not globally defined. In this sense the field configurations $ \phi^a \to^{r\to \infty} \frac{v r^a}{|r|}$ and $ \phi^a = (0,0,v)$ are truly distinct and describe different physical phenomena.
 +
 +By inspecting the Hedgehog field configuration further, one can see that it is the source of a Coulomb like potential that looks exactly like the potential of a Dirac monopole.
 +
 +The similarities between this topological non-trivial scalar field configuration and the Dirac monopole go even further.
 +
 +1.) Recall that we investigate a model where an $SU(2)$ gauge symmetry breaks to the abelian subgroup $U(1)$. At spatial infinity, the scalar field takes only values that minimize the scalar potential and thus there the gauge symmetry is only the unbroken $U(1)$ symmetry. The $SU(2)$ symmetry gets broken by the non-zero vev. Now one similarity is that the Dirac monopole appeared in a $U(1)$ pure gauge theory. One says that outside the "​monopole core" the Dirac and the 't Hooft-Polyakov monopole are indistinguishable.
 +
 +2.) We are able to bring the Hedgehog field configuration into the unitary gauge, where the scalar field points in the same direction everywhere, except for the monopole. There, the gauge transformation that achieves this "​combing"​ of the Hedgehog is singular. This singularity is exactly the same singularity that we call the Dirac string for the Dirac monopole. One can show (see Shnir page 154-155) that in the unitary gauge, the monopole potential is exactly the Dirac potential embedded in an $SU(2)$ group.
 +
 +Now this can be really confusing because the Dirac monopole appears in a pure $U(1)$ gauge theory, whereas for the 't Hooft-Polyakov monopole, so far, we were mainly concerned with the configuration of a scalar field.
 +
 +The thing is that for the t Hooft-Polyakov monopole, we actually have the freedom to shift our perspective. By performing the singular gauge transformation that combs the Hedgehog scalar field configuration (except at the south pole), the scalar field configuration at infinity becomes trivial. However, this gauge transformation has an effect on the other parts of the theory, too!
 +
 +This can be best understood by defining a gauge-invariant electromagnetic field strength tensor (see: Topology of Higgs fields ​ by J. Arafune, P. G. O. Freund, and C. J. Goebel)
 +
 +$$ F_{\mu\nu} = \partial_\mu A_\nu -\partial_nu A_\mu + \frac{1}{v^3e} \epsilon_{abc} \phi^a \partial_mu \phi^b \partial_\nu \phi^c .$$
 +
 +In the unitary gauge, where we have "​combed"​ the Hedgehog, the scalar field configuration at spatial infinity is trivial and thus carries winding number $0$. However, now the gauge field configuration becomes non-trivial and suddenly carries winding number $1$. The singular gauge transformation (eq. 5.3 in Shnir) that brings the scalar field into the unitary gauge "​transfers"​ the magnetic charge from the scalar field to the gauge field!
 +
 +Now, we want to understand this situation using again notions from topology. The scalar field at spatial infinity defines a map $S^2 \to S^2_{vac}$. Such maps are defined by the second homotopy class $\pi_2(G/​H)$,​ which in our case is $\pi_2(SU(2)/​U(1))= \pi_1(S^2_{vac})= \mathbb{Z}$.
 +
 +In contrast, a Dirac monopole was characterized by the first homotopy class of the gauge group $\pi_1(U(1))$.
 +
 +The clou is now that  $\pi_2(G/​H)$ and $\pi_1(H)$ are isomorphic!!!
 +
 +We can't rotate the Hedgehog field configuration smoothly to the trivial configuration $\phi_0$ globally on the sphere at infinity $S^2$. However, we can transform it into the $\phi$ on the two hemispheres separately. To get a consistent description this way, we need to define a transition function in the area where the two descriptions overlap, i.e. , for example, at the equator. (It doesn'​t matter where we choose that our descriptions (our two patches) overlap).
 +
 +By definition, the scalar field is everywhere on $S^2$ in a configuration that minimizes the scalar potential and thus the gauge symmetry is broken here from $SU(2)$ to $U(1)$. Therefore, the gauge transformation that transforms the description on one hemisphere into the description on the other hemisphere, must be an element of the unbroken subgroup $U(1)$.
 +
 +The scalar field yields a field value for each point on the sphere at spatial infinity, i.e. is a map from $S^2$ to the vacuum manifold. By combining this fact, with the patched description and the transition function idea from above, we get a correspondence between the map from $S^2$ to $S^2_{vac}= G/H$ and the map from $S^1$, (the equator) to $U(1)$. It can be shown that this map is an isomorphism (See Shnir page 167)!
 +
 +The isomorphism $\pi_2(G/H) = \pi_1(H)$ means that the topological charge that we used to characterize the Hedgehog scalar field configuration is equivalent to the topological charge that we use the characterize a Dirac monopole!
 +
 +----
  
 <​blockquote>​ <​blockquote>​
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 </​blockquote>​ </​blockquote>​
  
-[{{ :​advanced_notions:​hedgehog.jpg?​nolink&​200 |Source: http://​bartholomewandrews.com/​projects/​Skyrmions.pdf}}]+[{{ :​advanced_notions:​hedgehog.jpg?​nolink&​200|Source:​ http://​bartholomewandrews.com/​projects/​Skyrmions.pdf}}] 
  
    
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   * [[http://​www.theory.caltech.edu/​~preskill/​pubs/​preskill-1984-monopoles.pdf|MAGNETIC MONOPOLES]] by John Preskil   * [[http://​www.theory.caltech.edu/​~preskill/​pubs/​preskill-1984-monopoles.pdf|MAGNETIC MONOPOLES]] by John Preskil
  
 +-----
  
 +
 +
 +<​blockquote>​
 +Ever since Dirac has proposed the Dirac monopole
 +generalizing the Maxwell’s theory, the monopole has become
 +an obsession theoretically and experimentally [1].
 +After the Dirac monopole we have had the Wu-Yang
 +monopole [2], the ’t Hooft-Polyakov monopole [3], the
 +grand unification (Dokos-Tomaras) monopole [4], and the
 +electroweak (Cho-Maison) monopole [5–9]. But none of
 +them except for the electroweak monopole might become
 +realistic enough to be discovered.
 +
 +<​cite>​https://​arxiv.org/​pdf/​1703.02188.pdf</​cite>​
 +</​blockquote>​
    
 <tabbox Researcher> ​ <tabbox Researcher> ​
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   ​   ​
-<tabbox Examples> ​ 
  
---> Example1# 
- 
-  
-<-- 
- 
---> Example2:# 
- 
-  
-<-- 
  
 <tabbox FAQ> ​ <tabbox FAQ> ​
advanced_notions/topological_defects/magnetic_monopoles.1513763781.txt.gz · Last modified: 2017/12/20 09:56 (external edit)