# A Is the Berry connection compatible with the metric?

#### vanhees71

Gold Member
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.
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

2018 Award
@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.

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#### 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.
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

2018 Award
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.

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#### 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?

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#### Demystifier

2018 Award
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?
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.

#### samalkhaiat

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?
"I prefer to pass".

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#### Jimster41

Gold Member
Aaaand, I'm still just baffled when someone talks about "closed-paths" in a physically real state space with "(t)" anywhere at all. Show me one single close path in space-time. I can imagine a closed path over time in space. But nothing ever goes back to t0.

And I know that Berry phase is not well defined for open-paths. But then I see the importance of "time ordering" in the derivation of Berry phase in the first place - the well-defined closed path case that is (p.32 of Quantum Field Theory of Many Body Systems" Wen) and I can't help but think as my old boss used to say "flip it" - is Berry phase describing quantitative enforcement of time ordering and simultaneity (via a “joint” or somehow de-formable or re-distributable Hamiltonian, some kind of "negotiated metric" in the place where QM experiments of Schwarzchild observers meet?

As the space-bound twin is traveling her ageing is slowing relative to her brother though she doesn't notice. Neither does he right. Whatever "time" is she is experiencing a different one than her brother somehow but the difference is in no-way noticeable to either one.

Aging is electro-chemical right, all about energy, the Hamiltonian topo-map - and... especially the persistent but elusive arrow of time also sometimes known as the "oh so strangely squishy and observer manipulable distribution of entropy in space-time " or “The annoyingly phenomenologically vague 2nd Law of thermodynamics”.

But both twins are aging electro-chemically, normally, according to physics that are identical in all those ways but para-metrically different (only) in ways that affect regular old thermodynamic electro-chemical aging (whatever time is - biological electro-chemical ageing is one thing it is). She is moving through Yang Mills Gauge parameter space right? Isn't that the space what notion of Berry curvature, phase and connection covers (or one of them) right. A space that is weird with respect to parallel transport on "closed-paths" or paths that de-parallelize then try to at least re-parallelize if not "close".

Only when they (the twins) get back together is the accumulated change striking - the work that slows her ageing is done by something that covers that parameter space and gives it metric, which Berry phase sort of describes.

So, doesn't something actually have to happen to her periodic table to manifest her slowed rate of aging? Something that covers that table entirely and perfectly in a way she couldn't possibly notice. Doesn't her periodic table move through the Yang-Mill's parameter space? What about her second-law? I mean how is real (not mathematically abstract) "time" defined without reference to the Second-Law?

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#### Demystifier

2018 Award
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 .

#### Jimster41

Gold Member
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 .
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.

But I got stuck right off because...

I just can't buy the idea that the "time reversibility of laws of physics" prove reality is in principle time reversible. Never have. Not trying to "break the laws of physics" just not sure that the "law of physics" are not an invention (one with profound irony) of the fully irreversible and 100% inescapable thermodynamic and macroscopic i.e."psychological" TA.

I'm a bit confused but at the same time understand your correction w/respect to the "space" of transport Berry phase refers to. I will work on that.

Still, show me a space of real (i.e. physical) parameters that are not subject to time evolution, that are not found in thermodynamic time. Even the ones our minds imagine (that you are about to type out) - taken to the necessary GR+QM limit are subject to your thermodynamic TA. How could they not be they only exist in our physical brains (or physical artifacts thereof - like books) at some macroscopic thermodynamic time and as such they are like everything physical entities subject to the same time evolution they protest and try to deny. Funny.

IOW we can imagine going backwards from the Cuachy surface mathematically but it is pure fantasy - induced dialectic, the required anti-thesis of natural reality. But not itself real.

Again, I'm not trying to break the laws of physics. I'm just wondering a lot if those laws are stuck in their own Platonic rut.

Having said that. Clearly Relativity tells us that time is "flexible" (hence the twin scenario and other experiments that prove time dilation is a physical property of nature). It is the most bizarre combination of flexible and inflexible...

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#### Jimster41

Gold Member
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.

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#### Jimster41

Gold Member
And even if you were making fun of me a little - I wholeheartedly agree.

And I am not advocating solipsism at all. Just trying understand. The cartoon by the way could be taken as decent Cliff Notes for Schrödinger’s elegant and shocking “What is Life”

What excites me about Berry Phase is that it seems to be a surprising result of trusting Newton’s rules for integration - mixed with some sweet trickery from Euler and quaternion i. And, it gives me a what feels like a concrete connection to cheatery of Yang Mills gauge solution for making physics GR invariant. Just re-parametrize the physics as you go... really.

So, does her (the rocket twin) Periodic table get smooshed somehow or not? Clearly her t (proper time) does. How’s that work chemically - in like her Mitochondria. The whole EM machinery has to know how much it is accelerating somehow.

Berry Phase?

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#### Jimster41

Gold Member
Love the "pime". Thanks for that link. Funny. Reminds me of my days back in Seminary. Clearly that comment section needed some Physics Forums (de)mentor energy. You were nice and still held a firm mainstream position. As a layperson that style is very much appreciated. Sometimes confusion and enthusiasm do get conflated with having a pet theory. I don't have pet theory, lots of pet questions clearly but I'm totally open to learning and don't think I "know".

Having said that... the row you were in about whatever that Moving Dimensions Theory was about did strike me as sounding like - the ADS-CFT thing?

And I also come down more on that side than on "block time". Free will at the end of the Day seemed to be Schroedinger's assertion at the end of "What is Life" and per your cartoon, of all the things that we should trust about what our senses tell us... that surely is one isn't it? If anything shouldn't the conversation with our senses about laws of physics without preference for direction of "pime" go more like,

senses: what are these physics without pime you speak of?
intellect: yes, you mean "time" - don't worry, it is something you couldn't possibly understand
senses:

I thought you were more onto something with the discussion of "entropy gradient distribution" in the first paper. To me it suggests enough of an ether to support a notion of QM space-time geometry, analogous to a substance with mechanical properties capable of supporting QM But something also "squishy" enough to support GR's non preference for frame, actual free will and some kind of thermodynamic causality enforcement consistent with our sensory experience even when a twin goes rocket-shipping and... you know doesn't age as much... wtf?!

I'm assuming that we have experiments (we do right?) that really bear out time dilation (as evidenced in pime). My understanding of GPS is that it wouldn't work without an accounting of time dilation (via approximations). So time dilation meets the standard of a physicality of pime - sensed, not just imagined.

I'll stop using "pime" now but I found your teaching from it helpful.

As I'm thinking about this stuff, re-reading "Deep Down Things" and wiki-ing I realize that the brilliance of Yang-Mills gauge approach and what the symmetry groups of the Standard Model represent is just finally sinking in. So I don't mean that the rocket twin's periodic table gets squished literally. I get that symmetry under acceleration (and all the other transformations) is the whole point of the Standard Model symmetry group representation. Just how does the Symmetry get enforced?

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#### vanhees71

Gold Member
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".

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#### Jimster41

Gold Member
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 .
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?

Anyway I’m pretty confused about why and how Berry phase is only well defined for closed paths even if I just grant the notion of closure w/respect to state vector path. Why can’t there be a Berry phase for the step right before the closure? If it’s a smooth differentiable state space doesn’t there have to be?

And I was mistaken in identifying time ordering with Berry Phase exactly. It was introduced with “Time ordered correlation functions... which can be calculated with path integration”

But then later he is saying that Berry Phase it represents “curvature” and he even states, “Both the EM field and Gravitational field are generalized Berry Phases. They describe the frustrations in parallel transportation for some more general vectors...” p47 “Quantum Field Theory of Many Body Systems”

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"Is the Berry connection compatible with the metric?"

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