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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?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.
Thanks for the reply!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.
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 @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.
No, but it seems to be consistent with another metric: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.
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.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.
Thank you for all your remarks.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.
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.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.
This really clears things up now. Thanks!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 metric and Berry connection live in the same space, which is already a connection (pun alert!) between them. Whether Berry connection is compatible with that metric is at least a meaningful question.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?
There's even a covariant derivative(slides 18, 19):This metric and Berry connection live in the same space, which is already a connection (pun alert!) between them. Whether Berry connection is compatible with that metric is at least a meaningful question.
"I prefer to pass".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?
fun paper. I will read the rest of it more carefully. I do enjoy your stuff and I appreciate your mixture of math and sincere philosophical exposition. I am fully expecting to learn something that I can buy and that is illuminating once I read it all the way through.Closed paths in the context of Berry phase are closed in the (parameter) space, not in space-time.
If you are interested in paths closed in space-time, see e.g. my http://de.arxiv.org/abs/gr-qc/0403121 .
About that, see also my https://fqxi.org/community/forum/topic/259"psychological" TA
Okay, you might be making fun of me... a little but that’s okay if I’m gonna learn something today. Love that first paper. Kind of exactly what I have been struggling with... the question then is how the gradient of macroscopic TA for the timelike observer on the curve gets “managed” over a GR quasi-loop (twin goes rocketing then comes back later) so that when the twins meet again the brother’s head doesn’t explode with sudden aging and the sister doesn’t feel anything weird when her aging gets... dilated. But then I guess that’s the missing understanding of microscopically GR ready QFT.About that, see also my https://fqxi.org/community/forum/topic/259
The irony is that the misunderstanding of the Swedish academy not to distinguish between "pime" and "time" (which is mostly due to Bergson) made them give the Nobel prize to Einstein explicitly NOT for his greatest achievement, i.e., General Relativity but ironically for the only of his famous discoveries that's outdated today, i.e., his light-quanta concept of "old quantum theory", which is corrected by modern QED, while GR still stands all the very comprehensive tests made since 1915, including the newest discoveries concerning gravitational waves and the "first photo of a black hole".About that, see also my https://fqxi.org/community/forum/topic/259
Well, without the senses in the first place the intellect's models will be just pure maths...
I’m confused by this though. Wen introduces Berry Phase first as emerging from the calculation of the Lagrangian for the path integral of the Propagator for a spin system. He is talking about integrating over a set of coherent states though. I get that it’s a path in state space. Okay, so it’s an open path in space time but a closed path in terms of traversing the set of coherent states. But then he also describes it as a feature of adiabatic evolution of a spinor in a constant magnetic field that changes orientation. Is that also an open path in time? I guess. I am pretty confused by how Time is always still one of the integrands and then the integration is considered to be over a “closed path” for what... all dimensions but t?Closed paths in the context of Berry phase are closed in the (parameter) space, not in space-time.
If you are interested in paths closed in space-time, see e.g. my http://de.arxiv.org/abs/gr-qc/0403121 .