Are Christoffel symbols measurable?

atyy

Gravity and Yang-Mills are gauge fields, but gravity is not like a Yang-Mills field in detail.

Matterwave

Electromagnetism is an "abelian" gauge field theory called Quantum Electro-dynamics. It is not a Yang-Mills theory. A Yang-Mills theory is a non-abelian gauge field theory like QCD or the Electro-weak theory.

There is no well established quantum field theory for gravity, so I'm not sure how you want us to answer the first part of your question.

waterfall

For gravity to be a force. It has to have gauge transformation equivalent. So we still don't know what it is and it is not the metric guv nor the Christoffel symbols. This is what you guys are saying, correct? (say yes for record purposes)

In essence, we don't know what part or what is the gauge representation of the graviton. But what's weird is this. Electroweak has 3 gauge bozons, strong force has 8. If gravity is part of a larger gauge group. Why does it only have one boson?

Maybe gravity is not really a force at all. Maybe it is pure geometry. Remember in GR there is no force of any kind. Just geometry. So if the AsD/CFT has a correlate in our world. Then GR is just a classical limit that equates to pure information in the AsD/CFT world that isn't based on force and geometry. Do you agree?

atyy

All the gauge fields are geometrical. This is what Matterwave was saying about a principal bundle in post #35.
Witten, The Problem Of Gauge Theory

waterfall

Thanks for this crucial idea. This was why I kept encountering the idea of fiber bundles when I studied the Maxwell Equations before and didn't know the connection. Thought they were proposing the Faraday field lines as fiber bundles. So this is also how Weyl united GR and EM by proposing a new 5th dimension which the String Theory took advantage of right now....

Matterwave

Kaluza was the one who proposed a fifth dimension on which the curvature gives you the Maxwell's equations. Klein later proposed a mechanism by which this fifth dimension could exist without us realizing it (compactification). Thus, this 5-D GR+E&M theory is called "Kaluza Klein theory". String theory uses ideas from this (extra dimensions, and compactification), but is not the same as this.

I don't know what Weyl has to do with that...

waterfall

Yes, checking the Elegant Universe book, it was Klein, not Weyl.

But what Weyl did was this http://www.ams.org/notices/200607/fea-marateck.pdf

"In a 1918 article Hermann Weyl tried to combine electromagnetism and gravity by requiring the theory to be invariant under a local scale change of the metric, i.e., gμν → gμν e^α(x), where x is a 4-vector. This attempt was unsuccessful and was criticized by Einstein for being inconsistent with observed physical results. It predicted that a vector parallel transported from point p to q would have a length that was path dependent. Similarly, the time interval between ticks of a clock would also depend on the path on which the clock was transported.
The article did, however, introduce

• the term “gauge invariance”; his term was Eichinvarianz. It refers to invariance under his scale
change. The first use of “gauge invariance” in English3 was in Weyl’s translation4 of his famous
1929 paper.
• the geometric interpretation of electromagnetism.
• the beginnings of nonabelian gauge theory. The similarity of Weyl’s theory to nonabelian gauge theory is more striking in his 1929 paper."

Objections?

DaleSpam

I am not so sure that all observables are scalars, but I am pretty sure that all observations are scalars.

twofish-quant

I don't think that is true. Imagine a variable phi(A) where A is not a point in space but rather a region. Phi is not going to transform as a scalar field. You can also have thermodynamic quantities which are undefined at a specific point and require averaging to have meaning.

Volume is an observable, but it's certainly not a scalar. Wealth is a defined observable, but it's not a scalar.

twofish-quant

This doesn't work. In most common situations, wealth has scaling symmetry but not translational symmetry. I.e. if you have an economic situation, you can describe that situation equivalently by multiplying it by a scaling factor (i.e. do all your calculations in euros rather than dollars). Economics is not translationally symmetric. (I.e. if you add a constant amount to an economic situation, you are describing a different situation).

This has a number of implications

1) the important quantities are log-price rather than price
2) debt and credits are invariant quantities. If A is in debt to B, we can describe the amount of debt equivalently in dollars and euros, but we cannot by a change of coordinates eliminate the debt

A lot of the equations of finance can be derived from gauge theory.

DaleSpam

In both cases an observation of phi(A) or the average thermodynamic quantities is the interaction of some measuring device with the system of interest in order to produce a number. That number is the same, regardless of the coordinate system used to describe the experiment. Therefore the observation is a scalar.

I think you are correct here, so I will modify my above statement:

I am pretty sure that not all observables are scalars, but I am pretty sure that all observations are scalars.

twofish-quant

No it's not. I have a can of Coke that is 16 fluid ounces in one coordinate system and 473.18 mL in another. That's not a scalar.

I think you might have to define what is an observation. One thing about fluid measurement is that it doesn't correspond to a quantum mechanical operator, so if you argue that fluids can't be "observed" in a field theoretic sense, then I might be inclined to agree with you, but that defines observed in a what that's different enough from the normal meaning of the word that one has to be careful.

One thing here is that narrowing "observations" to things that can only be described in terms of fields is much too heavy a restriction. I take a coke can, fill it with water, and then dump out the water into a bucket of known volume. That doesn't fit well in field theory. For that matter prices are observations, but they don't fit into field theory and they certainly are not scalars (i.e. an observation of price gets you different numbers based on whether you are talking about dollars or euros).

PAllen

That's units, not an issue of invariance. By that standard nothing is invariant. The norm of proper acceleration (for example) is a scalar if anything is, but it still has units that are purely conventional.

twofish-quant

Another observable that's not a scalar. Color. In order to specify color you need to include three components (R, G, B). If you have only one component, you've measured "redness", "greenness" or "blueness' but you haven't measured color. Also because of redshift, different observers in different coordinate systems will see different colors, and different people will see different colors in quantifiable and predictable ways (i.e. if you are color blind, the coordinate system changes).

Now you could argue that all observables can be decomposed into scalars, but that's something quite different.

twofish-quant

One way of thinking about differential geometry in terms of units of measure. Also I'm using the term "scalar" in a very mathematically narrow sense, and "observation" in a very broad sense. The reason I do that is that I can open a book on differential geometry and get a mathematically precise definition of "scalar" whereas there isn't an obvious mathematically precise definition for "observation."

I think the question is "invariant with respect to what." In relativity, you typically keep thermodynamic quantities invariant, and then consider only coordinate transforms in space-time.

If you use the "tight" definition of scalar, then clearly it's not. Also, you **can** tell what's a scalar quantity by looking at the units. The fine-structure constant and pi are scalar with respect to everything. Rest mass is a scalar quantity. Charge is a scalar quantity.

This sounds like a massive nitpick. It is, but if you make these very fine distinctions then all sorts of useful things happen.

twofish-quant

I think we are have different definitions of what a "scalar" is. I'm defining it as a quantity that doesn't change when you change coordinate systems. I measure something in my coordinate system. You measure something in your coordinate system. We get the same number. There are some things that you can measure that have that characteristic (electric charge if you vary only space and time coordinates). There are some things that you can measure that *don't* have that characteristic (volume). Classifying things according to how they behave turns out to be useful.

DaleSpam

Yes, it would be important to define what is an "observation" and also what is an "observable". I have a feeling that the disagreement in this thread is primarily due to poor definitions.

I think that Matterwave et al. are talking about "observations" being scalars and I think that ApplePion et al. are talking about "observables" not being scalars. And I think that the disagreement is that they are using the same words for two different concepts.