Soliton v.s. homotopy group

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Main Question or Discussion Point

I don't understand the link from soliton solution of QFT to the homotopy group.
The argument goes like following:

Consider the field configuration such that the action is finite,
therefore we must require the field vanishes at spacetime infinity,
hence, we defined a map from the spacetime manifold to the group manifold of the theory.

I don't understand the last sentence of my previous paragraph.
Let me quote an example from Weinberg's QFT book of volume II.
He considered the Goldstone boson part of action in [tex] d [/tex]-dim Euclidean space(d>2),
[tex] S[\pi] = \int d^dx \left[ \frac{1}{2}\sum_{a,b} g_{ab}(\pi) \partial_i \pi_a \partial_i \pi_b + \cdots \right] [/tex]
To have finite action [tex] S [/tex], we impose the boundary condition that
[tex] \partial_i \pi_a \rightarrow 0[/tex] at spacetime infinity.
He then said something I don't understand:

"The Goldstone boson field [tex] \pi_a [/tex] at any point form a homogeneous space, the coset space [tex]G/H[/tex], for which it is possible to transform any one field value to any other by a transformation of [tex]G[/tex], so by a global [tex]G[/tex] transformation it is always possible to arrange that the asymptotic limit [tex]\pi_{a\infty}[/tex] takes any specific value, say 0"

Then, he made a conclusion which I don't understand how he got this:

"The filed [tex] \pi_a [/tex] thus represents a mapping of the whole [tex]d[/tex]-dimensional space, with the sphere [tex]r=\infty[/tex] taken as a single point, into the manifold [tex]G/H[/tex] of all field values."

So, after the emergence of this mapping, one could classify it according to homotopy group.
But I don't understand why such a "finite" action consideration leads to such a mapping between two manifolds, i.e. spacetime manifold(with infinity taken as a point) and group manifold.

Any discussion would be appreciated.
 

Answers and Replies

  • #2
160
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I don't understand the link from soliton solution of QFT to the homotopy group.
The argument goes like following:

Consider the field configuration such that the action is finite,
therefore we must require the field vanishes at spacetime infinity,
hence, we defined a map from the spacetime manifold to the group manifold of the theory.

I don't understand the last sentence of my previous paragraph.
Let me quote an example from Weinberg's QFT book of volume II.
He considered the Goldstone boson part of action in [tex] d [/tex]-dim Euclidean space(d>2),
[tex] S[\pi] = \int d^dx \left[ \frac{1}{2}\sum_{a,b} g_{ab}(\pi) \partial_i \pi_a \partial_i \pi_b + \cdots \right] [/tex]
To have finite action [tex] S [/tex], we impose the boundary condition that
[tex] \partial_i \pi_a \rightarrow 0[/tex] at spacetime infinity.
He then said something I don't understand:

"The Goldstone boson field [tex] \pi_a [/tex] at any point form a homogeneous space, the coset space [tex]G/H[/tex], for which it is possible to transform any one field value to any other by a transformation of [tex]G[/tex], so by a global [tex]G[/tex] transformation it is always possible to arrange that the asymptotic limit [tex]\pi_{a\infty}[/tex] takes any specific value, say 0"

Then, he made a conclusion which I don't understand how he got this:

"The filed [tex] \pi_a [/tex] thus represents a mapping of the whole [tex]d[/tex]-dimensional space, with the sphere [tex]r=\infty[/tex] taken as a single point, into the manifold [tex]G/H[/tex] of all field values."

So, after the emergence of this mapping, one could classify it according to homotopy group.
But I don't understand why such a "finite" action consideration leads to such a mapping between two manifolds, i.e. spacetime manifold(with infinity taken as a point) and group manifold.

Any discussion would be appreciated.
I found another example.
For 4D Euclidean, say, SU(n) gauge theory,
if we look for finite action field configuration,
the asymptotic form of gauge potential would be [tex] A(x) \sim g(x)\partial g^{-1} [/tex]
, i.e. in the pure gauge.
In this way, for a spacetime point [tex] x [/tex] at infinity, we have a group element [tex]g(x)[/tex], so, we defined a mapping from the "boundary" of the spacetime (in this case S^3) to the group manifold (if we consider SU(n), it's also S^3).

In this instanton case, the mapping between spacetime manifold (in this case, not full spacetime, but the boundary S^3) and group manifold is defined via the function [tex] g(x) [/tex].

But, then I have a question.
Suppose given a SU(2) gauge theory,
I don't even need to consider the field configuration such that the action is finite,
I already have a mapping from the full spacetime to the group manifold, i.e.
I have already this function [tex] g(x) [/tex], where [tex] g[/tex] is a group element and [tex] x[/tex] is any spacetime point.
That is, we automatically get a mapping from full spacetime to the group manifold in the very beginning of a gauge theory!

However, why, why nobody cares about such a mapping?
could anybody explain this for me?
thanks
 
  • #3
649
2
I found another example.
For 4D Euclidean, say, SU(n) gauge theory,
if we look for finite action field configuration,
the asymptotic form of gauge potential would be [tex] A(x) \sim g(x)\partial g^{-1} [/tex]
, i.e. in the pure gauge.
In this way, for a spacetime point [tex] x [/tex] at infinity, we have a group element [tex]g(x)[/tex], so, we defined a mapping from the "boundary" of the spacetime (in this case S^3) to the group manifold (if we consider SU(n), it's also S^3).

In this instanton case, the mapping between spacetime manifold (in this case, not full spacetime, but the boundary S^3) and group manifold is defined via the function [tex] g(x) [/tex].

But, then I have a question.
Suppose given a SU(2) gauge theory,
I don't even need to consider the field configuration such that the action is finite,
I already have a mapping from the full spacetime to the group manifold, i.e.
I have already this function [tex] g(x) [/tex], where [tex] g[/tex] is a group element and [tex] x[/tex] is any spacetime point.
That is, we automatically get a mapping from full spacetime to the group manifold in the very beginning of a gauge theory!

However, why, why nobody cares about such a mapping?
could anybody explain this for me?
thanks
The important part regarding the behaviour at infinity (on the S^3) is that the system cannot change continuously between states where the asymptotic behaviour of g(x) at large x have different homotopy classes, without having to cross a state with infinite energy.

On the other hand, within any finite volume in 4-space, the system can evolve from any g(x) to any other without needing infinite energy. So the behaviour of g(x) within any bounded domain is not topologically significant in the same way.

So the configuration at infinity is the topological invariant that can be used to classify the different states.
 

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