# A 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!

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#### Demystifier

2018 Award
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

2018 Award
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

Gold Member
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),

#### Demystifier

2018 Award
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

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

#### Demystifier

2018 Award
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

2018 Award
I'm still puzzled about to which "metric" you are referring to in the case of E&M.
4-dimensional spacetime metric.

#### vanhees71

Gold Member
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

2018 Award
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

2018 Award
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

Gold Member
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

2018 Award
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 Geometry/Eguchi, Gilkey, Hanson - Gravitation, Gauge Theories And Differential Geometry (Pr 1980).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

Gold Member
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

2018 Award
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

Gold Member
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) :)

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#### vanhees71

Gold Member
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

2018 Award
@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

"Is the Berry connection compatible with the metric?"

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