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Are Christoffel symbols measurable? 
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#55
Feb1612, 07:32 PM

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Witten, The Problem Of Gauge Theory 


#56
Feb1612, 08:02 PM

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#57
Feb1612, 08:28 PM

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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 5D 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... 


#58
Feb1612, 08:38 PM

P: 381

But what Weyl did was this http://www.ams.org/notices/200607/feamarateck.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 4vector. 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? 


#59
Feb1612, 11:54 PM

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



#60
Feb1712, 05:16 AM

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


#61
Feb1712, 05:29 AM

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This has a number of implications 1) the important quantities are logprice 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. 


#62
Feb1712, 06:41 AM

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


#63
Feb1712, 09:23 AM

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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). 


#64
Feb1712, 09:29 AM

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PF Gold
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#65
Feb1712, 09:37 AM

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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. 


#66
Feb1712, 09:50 AM

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


#67
Feb1712, 09:59 AM

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#68
Feb1712, 10:01 AM

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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. 


#69
Feb1712, 10:45 AM

P: 6,863

Also I think that I've thought of an observable that clearly is not a scalar. Chriality. A particle is either lefthanded or righthanded, and since this is a binary quantity. It's not a scalar.
For that matter, you flip a coin, the "headness" or "tailness" of the coin is a boolean quality which is not a scalar. For that matter any observation or observable that is binary isn't a scalar. Finally, for the people that still insist that observations have to be a scalar, how do you know it's a scalar and not a pseudoscalar? I have a feeling that "height" is a scalar, but "leftness" is a pseudoscalar. 


#70
Feb1712, 11:05 AM

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However, an observation of any of the quantities you have mentioned is a scalar (i.e. it is unchanged under diffeomorphisms) even if the corresponding observable is not. 


#71
Feb1712, 11:16 AM

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Once you have measured something the observation is a scalar. If you perform some experiment and the number 7.43 pops out on your measuring device then no change of coordinate systems can possibly change that number to anything other than 7.43. Therefore, the number measured is a scalar. It may be that you claim that 7.43 is a length and I disagree, but regardless of how we interpret the number in terms of physical quantities in our favorite coordinate system, we will agree that the number is the same. That makes it a scalar. 


#72
Feb1712, 12:28 PM

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PF Gold
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Note, volume is trivially a scalar  it is integral of volume element, which is differential contraction of the metric. It is just as much a scalar invariant as proper time. 


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