Is the Berry connection compatible with the metric?

[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?

 

[Demystifier]

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.

 

[Joker93]

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.

There's even a covariant derivative(slides 18, 19):
https://www.google.com.cy/url?sa=t&...XDHQQFggkMAE&usg=AOvVaw1NZsBx_hvXTOBdFhvwexhy

 

[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".

 

[Jimster41]

"I prefer to pass".

Only for a periodic metric...?

 

[Jimster41]

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 Schwarzschild 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?

 

[Demystifier]

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]

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...

 

[Demystifier]

@Jimster41 your ideas remind me of

 

[Demystifier]

"psychological" TA

About that, see also my https://fqxi.org/community/forum/topic/259

 

[Jimster41]

About that, see also my https://fqxi.org/community/forum/topic/259

Okay, you might be making fun of me... a little but that’s okay if I’m going to 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.

 

[Jimster41]

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?

 

[Jimster41]

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?

 

[vanhees71]

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".

 

[MathematicalPhysicist]

@Jimster41 your ideas remind me of
View attachment 248533

Well, without the senses in the first place the intellect's models will be just pure maths...

 

[Jimster41]

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”