Is the Berry connection compatible with the metric?

[Joker93]

Hello,
Is the Berry connection compatible with the metric(covariant derivative of metric vanishes) in the same way that the Levi-Civita connection is compatible with the metric(as in Riemannanian Geometry and General Relativity)?

Also, does it have torsion? It must either have torsion or not be compatible with the metric, otherwise it is a Levi-Civita connection, which I think it is not.

Thank you!

 

[Demystifier]

Berry connection is more like electromagnetic gauge connection $A_{\mu}$, so it is not related to metric.

 

[Joker93]

Berry connection is more like electromagnetic gauge connection $A_{\mu}$, so it is not related to metric.

But there is a metric in the theory of Berry phase. It arises from what is called the quantum geometric tensor. It can be found in the article by Berry called "The Quantum Phase, 5 years after".
Also, the compatibility plays a role on how we take derivatives, i.e. if
D(<m|n>)=<Dm|n>+<m|Dn> is true, where D denotes the covariant derivative.

 

[Demystifier]

But there is a metric in the theory of Berry phase. It arises from what is called the quantum geometric tensor. It can be found in the article by Berry called "The Quantum Phase, 5 years after".
Also, the compatibility plays a role on how we take derivatives, i.e. if
D(<m|n>)=<Dm|n>+<m|Dn> is true, where D denotes the covariant derivative.

https://michaelberryphysics.files.wordpress.com/2013/07/berry187.pdf
I would say it's analogous to the fact that electrodynamics is also defined in a space with metric, despite the fact that the electromagnetic connection is not related to that metric.

 

[vanhees71]

It's not clear to me what you mean concerning E&M. Isn't this related to the AB effect and the very elegant and beautiful differential-geometric approach to its derivation, according to

T. T. Wu and C. N. Yang, Concept of nonintegrable phase factors and global formulation of gauge fields, Phys. Rev. D 12, 3845 (1975),
http://link.aps.org/abstract/PRD/v12/i12/p3845 ?

 

[Joker93]

https://michaelberryphysics.files.wordpress.com/2013/07/berry187.pdf
I would say it's analogous to the fact that electrodynamics is also defined in a space with metric, despite the fact that the electromagnetic connection is not related to that metric.

So, D(<m|n>)=<Dm|n>+<m|Dn> which has to do with metric compatibillity is not true?

 

[Demystifier]

It's not clear to me what you mean concerning E&M.

In both cases the gauge group is U(1). And the metric is not a metric of the U(1) manifold.

 

[vanhees71]

I'm still puzzled about to which "metric" you are referring to in the case of E&M.

 

[Demystifier]

So, D(<m|n>)=<Dm|n>+<m|Dn> which has to do with metric compatibillity is not true?

The equation is true, but that equation does not express the metric compatibility. Metric compatibility is the statement that covariant derivative of the metric vanishes.

 

[Demystifier]

I'm still puzzled about to which "metric" you are referring to in the case of E&M.

4-dimensional spacetime metric.

 

[vanhees71]

Ok, but what has this to do with the Berry phase or the nonintegrable phase factor in E&M?

 

[Joker93]

The equation is true, but that equation does not express the metric compatibility. Metric compatibility is the statement that covariant derivative of the metric vanishes.

But that's what the mathematician's call it.
It's the same thing. It's like the product rule where the derivative of the metric vanishes.
You can see Do Carmo's (Riemannian geometry) definition in his chapter on affine connections where he defines the compatibility with the metric.

 

[Demystifier]

Ok, but what has this to do with the Berry phase or the nonintegrable phase factor in E&M?

Just an analogy, see #7.

 

[Demystifier]

But that's what the mathematician's call it.
It's the same thing. It's like the product rule where the derivative of the metric vanishes.
You can see Do Carmo's (Riemannian geometry) definition in his chapter on affine connections where he defines the compatibility with the metric.

I am not sufficiently fluent in mathematician's terminology, perhaps @samalkhaiat may help. From a physical perspective, I would advise you to ask yourself the following related question. In electrodynamics, is the covariant derivative
$$\nabla_{\mu}=\partial_{\mu}-ieA_{\mu}$$
metric compatible?

 

[vanhees71]

This doesn't make sense since there's no metric here (the GR metric doesn't count here since it hasn't anything to do with em. gauge). The gauge-covariant derivative is a connection (with the gauge field the analogue of Christoffel symbols and the field-strength tensor $\propto [\nabla_{\mu},\nabla_{\nu}]$ as the analogue of the curvature.

 

[Demystifier]

This doesn't make sense since there's no metric here (the GR metric doesn't count here since it hasn't anything to do with em. gauge). The gauge-covariant derivative is a connection (with the gauge field the analogue of Christoffel symbols and the field-strength tensor $\propto [\nabla_{\mu},\nabla_{\nu}]$ as the analogue of the curvature.

Yes, that's exactly what I said in some of the posts above, using a slightly different words.

Or let me try to express it in more mathematical language (which OP seems to prefer), with a caveat that I am not so fluent in mathematical terminology so I cannot guarantee that what I will say is fully correct. As can be seen e.g. in the review http://xavirivas.com/cloud/Differential%20Geometry/Eguchi,%20Gilkey,%20Hanson%20-%20Gravitation,%20Gauge%20Theories%20And%20Differential%20Geometry%20(Pr%201980).pdf
Sec. 5, one should distinguish connections on tangent bundles from connections on principal bundles. @Joker93 seems to try express everything in terms of connections on tangent bundles, while the right language in the context of Berry connection seems to be the language of connections on principal bundles. This is what his source of confusion seems to be.

 

[vanhees71]

I'm also not fluent in the "jet-bundle language", but as far as I understand it indeed in differential geometry of differentiable manifolds you usually consider connections on tangent bundles (which also doesn't need a metric or pseudo-metric or even a (pseudo-)Riemannian space), while the gauge-covariant derivatives define connections on principal bundles. For a physicist's naive approach to non-Abelian gauge theories (everything of course also applies to the Abelian case) from a geometrical perspective, see the beginning of Chpt. 7 in

https://th.physik.uni-frankfurt.de/~hees/publ/lect.pdf

 

[Demystifier]

I'm also not fluent in the "jet-bundle language", but as far as I understand it indeed in differential geometry of differentiable manifolds you usually consider connections on tangent bundles (which also doesn't need a metric or pseudo-metric or even a (pseudo-)Riemannian space), while the gauge-covariant derivatives define connections on principal bundles. For a physicist's naive approach to non-Abelian gauge theories (everything of course also applies to the Abelian case) from a geometrical perspective, see the beginning of Chpt. 7 in

https://th.physik.uni-frankfurt.de/~hees/publ/lect.pdf

Nice to see that we agree again. (We agree on almost everything, except on interpretations of QM and value of Zee's QFT. )

 

[vanhees71]

Well, if everybody agrees with everybody you won't learn anything ever in the forums. Fortunately that's not the case, and at least I learn tons of new interesting things here!

 

[Joker93]

I am not sufficiently fluent in mathematician's terminology, perhaps @samalkhaiat may help. From a physical perspective, I would advise you to ask yourself the following related question. In electrodynamics, is the covariant derivative
$$\nabla_{\mu}=\partial_{\mu}-ieA_{\mu}$$
metric compatible?

The covariant derivative of EM is compatible with the spacetime metric. This can be seen by the definition given by Nakahara in his textbook Geometry, Tpology and Physics, p.253(see image below). You can see that my definition and metric being covariantly constant are equal. The metric compatibility condition is not exclusive to the Levi-Civita connection. In the EM case, the covariant derivative is compatible with the spacetime metric, but it is not the Levi-Civita connection that performs parallel transports on the spacetime manifold. And the compatibility with the metric need not refer to the Levi-Civita connection. It is a property that any connection can have, either L-C or not.
Please correct me if I am misunderstanding something here.

Now, in the case on the Berry connection, since I am always using $$\partial_{\mu}(<n|m>)=<\partial_{\mu}n|m>+<n|\partial_{\mu}m>$$, I think that the Berry connection is compatible with the metric(since the covariant derivative is given by $$\nabla_{\mu}=\partial_{\mu}-ieA_{\mu}$$ as you said), although not the spacetime metric. I suspect that the metric that is relevant to the Berry connection case is that which is used to define the inner product between states; I mean, since we do have a well-defined inner product between states, then there should be a metric behind it.

Lastly, note that there exists some kind of metric that comes from what is called the Quantum Geometric tensor.

@Demystifier you are correct in that I am not so fluent in the language of bundles, so I am effectively(though not purposely) trying to convert my understanding of the Berry language to the language used in Riemannian geometry(which I know well, at least at an introductory level). :)

@vanhees71 I am tagging you too so you can give an opinion on what I am saying(if you don't mind, that is) :)

 

[vanhees71]

I don't understand the statement "In the EM case, the covariant derivative is compatible with the spacetime metric". I don't see what it has to do with the spacetime metric at all. In some hand-waving way one could say that the covariant derivative lives in the tangent space of the Lie group, but not in tangent space of the spacetime manifold. I guess mathematicians won't like this hand-wavy argument at all.

 

[Joker93]

I don't understand the statement "In the EM case, the covariant derivative is compatible with the spacetime metric". I don't see what it has to do with the spacetime metric at all. In some hand-waving way one could say that the covariant derivative lives in the tangent space of the Lie group, but not in tangent space of the spacetime manifold. I guess mathematicians won't like this hand-wavy argument at all.

What I am saying is that we can talk about metric compatibility no matte what the metric or the connection is. I am just taking the definition given by Nakahara(see edited previous post). So, indeed it does not have to do only with the spacetime metric.
So, in essence, I am saying that we can say that a connection is compatible with a metric(not the metric) if the derivative of the inner product is given by the product rule as given by Nakahara.
I can't quite understand the language of bundles yet, so forgive me for pushing it a bit.

 

[Demystifier]

@Demystifier you are correct in that I am not so fluent in the language of bundles, so I am effectively(though not purposely) trying to convert my understanding of the Berry language to the language used in Riemannian geometry(which I know well, at least at an introductory level). :)

The geometry of principal bundles is not Riemannian geometry. Or in physical words, the geometry of gauge theories and Berry connections is not Riemannian geometry. You are trying to do an impossible conversion.

 

[Joker93]

The geometry of principal bundles is not Riemannian geometry. Or in physical words, the geometry of gauge theories and Berry connections is not Riemannian geometry.

Of course not, and that is the reason that I am trying to understand what you are saying, and sorry for pushing it.
What I am asking, in essence, is whether or not we can define metric compatibility outside of Riemannian geometry(where we are concerned with the tangent bundle). I thought that the definition of metric compatibility does not have to do with the bundle considered but with the relationship between the given connection with the given metric. I mean, the definition does not seem to be concerned with the bundle into consideration.

 

[Joker93]

@Demystifier @vanhees71
Here: https://en.wikipedia.org/wiki/Metric_connection#Metric_compatibility
In the definition of metric compatibility, there does not seem to be a restriction that we are talking about the tangent bundle. That case(tangent bundle) is examined in the previous section of Riemannian geometry.

EDIT: Also, the definition (https://en.wikipedia.org/wiki/Metric_connection#Definition) does not refer to a specific bundle.

 

[vanhees71]

Sure, that's the usual definition of a connection compatible with the metric of a Riemannian space, but there's no metric related with the gauge-covariant derivative it could be compatible with.

 

[Joker93]

Sure, that's the usual definition of a connection compatible with the metric of a Riemannian space, but there's no metric related with the gauge-covariant derivative it could be compatible with.

But, since there is the inner product between quantum states, then couldn't we find, at least in principle, the corresponding(to the inner product) metric?

 

[romsofia]

I've, personally, never heard of the Berry connection. Are you asking if the Berry Connection is torsion free, or is not symmetric in the christoffel symbols?

It sounds like you're asking if this connection is coupled to Riemannian geometry? I'm also confused as the others because, in principle, to me you're asking something along the lines of the minimal coupling principle. So i'll outline the steps!

If you want to couple something to Riemannian geometry, essentially, what you do is:
1) Replace $n_{\mu\nu}$ with $g_{\mu\nu}$
2) Replace patrial/total derivatives with covariant derivaties
3) use $\sqrt{-g}$ to saturate all tensor densities to zero. If you're in an action integral, you've probably seen this as $d^4x$ becoming $d^4x\sqrt{-g}$

You can try this out with the E+M tensor $F_{\mu\nu} \rightarrow \nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu}$ and you'll see the symmetry of the christoffel symbols here.

I'd also suggest learning about differential forms if you have not, it makes this whole concept of tangent spaces and all that easier to swallow (IMO). If I'm going down the right trail here, let me know and I'll try to add on more about this principle.

 

[Joker93]

I've, personally, never heard of the Berry connection. Are you asking if the Berry Connection is torsion free, or is not symmetric in the christoffel symbols?

It sounds like you're asking if this connection is coupled to Riemannian geometry? I'm also confused as the others because, in principle, to me you're asking something along the lines of the minimal coupling principle. So i'll outline the steps!

If you want to couple something to Riemannian geometry, essentially, what you do is:
1) Replace $n_{\mu\nu}$ with $g_{\mu\nu}$
2) Replace patrial/total derivatives with covariant derivaties
3) use $\sqrt{-g}$ to saturate all tensor densities to zero. If you're in an action integral, you've probably seen this as $d^4x$ becoming $d^4x\sqrt{-g}$

You can try this out with the E+M tensor $F_{\mu\nu} \rightarrow \nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu}$ and you'll see the symmetry of the christoffel symbols here.

I'd also suggest learning about differential forms if you have not, it makes this whole concept of tangent spaces and all that easier to swallow (IMO). If I'm going down the right trail here, let me know and I'll try to add on more about this principle.

Thanks for the reply!
I am not talking about minimal coupling, nor does my question have to do with the spacetime metric in particular.
The Berry connection is like the 4-potential in EM and the Berry curvature is like the EM tensor, but you must replace the spacetime derivatives with derivatives eith respect to certain parameters. It is a problem in QM and does not have to do with Riemannian geometry(which has to do with the tangent bundle), but it has to do with the principal bundle(as others pointed out).
What I am asking though is about metric compatibility(which I think that a physicist sees for first time in General Relativity).
It is defined as the connection for which the following is true:
D(<m|n>)=<Dm|n>+<m|Dn> where <m|n> is the inner product between states and D is the covariant derivative associated with the connection that we are considering.
I the picture in post 20(Nakahara) you can see that it is equivalently defined as the vanishing of the covariant derivative of the metric that is associated with that inner product.
So, in essence, what I am asking is whether or not the Berry connection is compatible with the metric associated with the inner product between quantum states.
I hope I did not confuse you.
Thank you!

 

[Demystifier]

@Demystifier @vanhees71
Here: https://en.wikipedia.org/wiki/Metric_connection#Metric_compatibility
In the definition of metric compatibility, there does not seem to be a restriction that we are talking about the tangent bundle. That case(tangent bundle) is examined in the previous section of Riemannian geometry.

EDIT: Also, the definition (https://en.wikipedia.org/wiki/Metric_connection#Definition) does not refer to a specific bundle.

Riemannian geometry can be defined without language of fiber bundles. But Riemannian geometry is not the most general geometry, and fiber bundles are introduced to describe more general geometries. In the language of fiber bundles, Riemannian geometry is a special case related to tangent bundles. As I said, you don't need to use any bundles at all to talk about Riemannian geometry, but then you cannot talk about non-Riemannian geometries such as geometry of Berry connection.

 

[Demystifier]

So, in essence, what I am asking is whether or not the Berry connection is compatible with the metric associated with the inner product between quantum states.

No, but it seems to be consistent with another metric:
https://physics.stackexchange.com/questions/116985/can-the-berry-connection-be-derived-from-a-metric

 

[Demystifier]

But that's what the mathematician's call it.
It's the same thing. It's like the product rule where the derivative of the metric vanishes.
You can see Do Carmo's (Riemannian geometry) definition in his chapter on affine connections where he defines the compatibility with the metric.

If you look at do Carmo's Riemannian Geometry, page 50, Definition 2.1, you will see that he defines affine connection as a map from $\chi(M)\times\chi(M)$ to $\chi(M)$, where $\chi(M)$ is a set of vector fields. In a physics-friendly language, this means that the affine connection is an object with 3 indices, all of which belong to the same space $\chi(M)$. Indeed, the Christoffel connection $\Gamma^{\mu}_{\alpha\beta}$ has 3 such indices. On the other hand, a Yang-Mills connection $A^{\mu}_{ab}$ has one spacetime index ${\mu}$ and two gauge group indices $a,b$, so it is not an affine connection. The electromagnetic connection $A^{\mu}\equiv A^{\mu}_{11}$ is a special case of Yang-Mills, where the gauge-group indices are trivial because they take only one value 1 as the gauge group $U(1)$ is 1-dimensional. The Berry connection is very much like $A^{\mu}$ because both lack (non-trivial) lower indices. In other words, Berry connection is not an affine connection.

 

[Joker93]

If you look at do Carmo's Riemannian Geometry, page 50, Definition 2.1, you will see that he defines affine connection as a map from $\chi(M)\times\chi(M)$ to $\chi(M)$, where $\chi(M)$ is a set of vector fields. In a physics-friendly language, this means that the affine connection is an object with 3 indices, all of which belong to the same space $\chi(M)$. Indeed, the Christoffel connection $\Gamma^{\mu}_{\alpha\beta}$ has 3 such indices. On the other hand, a Yang-Mills connection $A^{\mu}_{ab}$ has one spacetime index ${\mu}$ and two gauge group indices $a,b$, so it is not an affine connection. The electromagnetic connection $A^{\mu}\equiv A^{\mu}_{11}$ is a special case of Yang-Mills, where the gauge-group indices are trivial because they take only one value 1 as the gauge group $U(1)$ is 1-dimensional. The Berry connection is very much like $A^{\mu}$ because both lack (non-trivial) lower indices. In other words, Berry connection is not an affine connection.

Thank you for all your remarks.
But, if you can check Nakahara on p.397, he defines a metric connection(metric compatible with a metric) as just a metric that preserves the inner product.
He does make a connection with Riemannian geometry through what he defines as Riemannian structure.
So, if I proceed on with this definition that has to do with fibre bundles, then the Berry connection is compatible with the metric that is used to take inner products between quantum states {|n>}, since the following is true:
$$\partial_\mu(<n|m>)=<\nabla_\mu n|m>+<n|\nabla_\mu m>$$ where $$\nabla_\mu=\partial_\mu+iA_\mu$$ where $A_\mu$ is the gauge field(or the Berry connection in this case).
I can't see anywhere in that section of Nakahara that restricts us to have affine connections in order for the definition he gives(for metric connections on fibre bundles) to be valid. Please correct me if I am wrong or I have missed something.

Also, on your post 31, I visited the link but I still can't figure out why we cannot say that the Berry connection is compatible with the the particular metric that has to do with the inner product between quantum states since the above is valid.

 

[Demystifier]

But, if you can check Nakahara on p.397, he defines a metric connection(metric compatible with a metric) as just a metric that preserves the inner product.

Nakahara defines the Riemannian structure on $E$. What you are missing is to ask yourself - what is $E$ in your case? Is it the quantum Hilbert space, or is it the parameter space associated with Berry connection? The metric you are talking about is a metric on the quantum Hilbert space. By contrast, the Berry connection is a connection on the parameter space. Therefore the metric in (10.73) cannot be interpreted as the metric in the quantum Hilbert space. If you want to talk about metric compatibility of the Berry connection, then you need another metric, a metric on the parameter space.

 

[Joker93]

Nakahara defines the Riemannian structure on $E$. What you are missing is to ask yourself - what is $E$ in your case? Is it the quantum Hilbert space, or is it the parameter space associated with Berry connection? The metric you are talking about is a metric on the quantum Hilbert space. By contrast, the Berry connection is a connection on the parameter space. Therefore the metric in (10.73) cannot be interpreted as the metric in the quantum Hilbert space. If you want to talk about metric compatibility of the Berry connection, then you need another metric, a metric on the parameter space.

This really clears things up now. Thanks!
Berry, in the article I attach below, talks about a quantum geometric tensor which measures distances in parameter space. Might it be that this metric has any connection with what we are discussing?