# A Is the Berry connection compatible with the metric?

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1. Jan 2, 2018

### 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!

2. Jan 2, 2018

### Demystifier

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

3. Jan 2, 2018

### Joker93

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.

4. Jan 2, 2018

### Demystifier

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.

5. Jan 2, 2018

### 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),

6. Jan 2, 2018

### Joker93

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

7. Jan 2, 2018

### Demystifier

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

8. Jan 2, 2018

### vanhees71

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

9. Jan 2, 2018

### Demystifier

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.

10. Jan 2, 2018

### Demystifier

4-dimensional spacetime metric.

11. Jan 2, 2018

### vanhees71

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

12. Jan 2, 2018

### Joker93

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.

13. Jan 3, 2018

### Demystifier

Just an analogy, see #7.

14. Jan 3, 2018

### Demystifier

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?

15. Jan 3, 2018

### 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.

16. Jan 3, 2018

### Demystifier

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.

17. Jan 3, 2018

### 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

18. Jan 3, 2018

### Demystifier

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

19. Jan 3, 2018

### 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!

20. Jan 3, 2018

### Joker93

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) :)

Last edited: Jan 3, 2018