In my hand, I have a can of Coke that is 16 fluid ounces in one coordinate system and 473.18 mL in another. Is that scalar or not? Answer: No. This is where the need to define observation comes to face (nh). One thing about fluid measurement is that it doesn’t correspond to any quantum mechanical operator so if you want to start an argument how fluids can’t be “observed” in the sense of fluid theoretics, I’ll be inclined to fall in agreement with you. However, please understand that a particular scenario such as the previously mentioned defines “observed” in what’s different enough from the normal meaning of the word. To narrow “observations” down to things that can only be described in terms of fields is pretty much too heavy of a restriction. With that same Coke can, I can fill it up with water and then commence to dumping that water out in a bucket of known volume. Take note that this does not fit well in field theory.
Another observable that is not scalar is: color. How would you go about specifying color? Well, you need three components (R, G, B). If you’re stuck with only one component, you can say that you’ve measured “red”, “green” or “blue” but you haven’t exactly measured color. What is color? Answer: Color is radiation–non-ionizing radiation, to be exact. In regards to redshift, different observers in different coordinate systems will see different colors and menageries of people will see different colors in quantifiable and predictable ways. In other words, the coordinate system will change for someone who is colorblind.
Do know that as I’m using the term scalar, it’s being used in a very mathematically narrow sense of the term and in contrast, the term “observation” is being used in an extremely broad sense of the term. For example, take differential geometry. One way of thinking about differential geometry would be in terms of units of measure. So, for a term such as scalar, I can open up a book on differential geometry and get a mathematically precise definition of scalar whereas there isn’t a mathematically precise definition for observation. 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, for example, is a scalar quantity as well as charge being a scalar quantity. Can you have different definitions for the term scalar? Well, in this instance, I’m defining scalar “…as a quantity that doesn’t change when you change coordinate systems”. If I were to measure something in my coordinate system and you were to measure something in your coordinate system, we both end up with the same number. There are things you can measure that have a particular characteristic such as an electric charge (provided that you’re varying only space and time coordinates); and there are things that you can measure that don’t have that particular characteristic, such as volume. You should always classify things in accordance to how they behave (observe/observing).
Now, for those of you who are “advanced”, can I ask you what do you think of chriality? Take some time to think of an observable that clearly isn’t scalar: a particle. A particle is either left-handed or right-handed. This would make a particle a binary quantity and a binary quantity is not scalar. Flip a coin, the quantity that’s considered the “head” and the quantity that’s considered the “tail” of the coin is a boolean quanity and boolean quantities are not scalar. To throw some salt in the wound, how do you know that a particular observation is a scalar and not a pseudo-scalar?
Some would say that measurements are always scalar. Personally, I think that one should be more concerned about how they perceive measurements to be and not from the experiments of physicists and elitists of the sorts. You can come up with a definition of “measurement” that will always give you something in return that’s scalar, however, it would definitely be a technical definition, by far, that wouldn’t have any obviously real relationship to either the common or actual definition nor have any bearing of a similarity towards the process of “measuring” or even “observing”. As I’ve said earlier, for color to specified, there’s a requirement of at least three numbers to do the specifying. Now, for someone to argue that “measurements” can only result in one number, specifically, then that would mean that color cannot be measured. Choosing whatever definition you would like to tie to measurement whenever you want to prove a theorem is fine but if the definition of measurement that you come up with negates color or colors (plural) then I, for one, am going to have a problem with that and that’s because the definition [of scalar] that I’m referring to ties-in with mathematics, which is different from the definition [of scalar] that’s looped-in with physics. The physics definition [of scalar] is inclusive of invariants under a subset of transformations whereas that restriction doesn’t exist in the definition that’s descriptive of the math. However, is there ever an event by consequence in which my take on the definition of scalar would deem that all observations are scalar? In terms of GR (general relativity), the answer would be a resounding “Yes”. In general relativity, something is a scalar if it is invariant to Lorentz rotations and translations and it matters not if the change is the result of scaling relationships. You aren’t confined to specific transformations in differential geometry.
Understand that the way I was using the definition of scalar does not relate to how it’s used in relativity, therefore, you can say that all observables are scalar if you are applying it towards the definition of scalar in relativity. In other words, number changes due to rotation and transformations do indeed matter. But outside of relativity, my former stance remains firm. I’m not convinced that only scalar quantities can be observed in contexts outside of relativity. You look at a wind tunnel with a wing and then you have little flags all pointing in different directions. It would appear that you are observing a vector field. With a weather pane pointing in the direction of the wind, it would seem that you are observing a vector. I say this because data visualization is basically you observing vector fields. If you were to tell an aerospace engineer that they aren’t really observing a vector but rather a collection of scalars, playa, you’ll get looked at as if you’re on some sort of mind-altering drug. When you apply differential geometry to either fluid dynamics, you’ll be observing vectors. Now, if you have a computational fluid dynamics flow that’s around a black hole with a field that describes the velocity of the fluid within a local reference frame of each corresponding point, it would be anyone’s guess that a person specialized in general relativity would describe the flow of the fluid as “scalar” since the components of the flow will not change when you do a Lorentz transform but the person specialized in computational fluid dynamics would describe the fluid flow as “vector” because he’s been convinced that one would be in need of multiple components to describe the field of velocity. What transpires after specifying a coordinate system? Answer: You still end up with something resembling a vector. If you were to do relativistic fluid dynamics, once you specify the reference frame, what you end up with is…….a vector. Fields of velocity make things complicated, playa, but color and composition form vector spaces that are in themselves, independent of the space-time vector spaces.
An extremely strong, philosophical statement that can be made is “…all measurements can be reduced to all measurements that physicists are used to making”. I consider a statement of that nature as false. Vector spaces and the math associated with it comes in handy when you have an underlying reality, such as color, that’s independent of the measurements themselves. As I’ve stated before, color requires three components to be specified but color is independent of those components. Color can be specified in terms of RGB or color temperature but color remains independent of those measurements. The existence of color is independent of those components, therefore, it’s a vector field. One could argue that scalars can be considered as abstractions. Take the time to measure light intensity; it goes into a meter that goes through your eyes into your brain, and what takes place? No one knows for certain. For someone to say that no vector field is directly observable, the problem that you’ll end up with is an extremely restrictive definition of the word “observe” by which it’s not clearly determined if anything is observable. Vector spaces are indeed useful, especially when it comes to representing things are that pure fantasy, I mean, any first person shooter video game will have vector space representation of all assortments of imaginary things.
and
…are both independent. So, when you’re deriving…
![[\phi(\vec{x},t),\dot{\phi}(\vec{x}',t)] = 0](http://s0.wp.com/latex.php?latex=%5B%5Cphi%28%5Cvec%7Bx%7D%2Ct%29%2C%5Cdot%7B%5Cphi%7D%28%5Cvec%7Bx%7D%27%2Ct%29%5D+%3D+0+&bg=ffffff&fg=808080&s=0 "[\phi(\vec{x},t),\dot{\phi}(\vec{x}',t)] = 0")
…I end up with terms like
![[\phi_1(\vec{x},t),\dot{\phi}_2(\vec{x}',t)]](http://s0.wp.com/latex.php?latex=%5B%5Cphi_1%28%5Cvec%7Bx%7D%2Ct%29%2C%5Cdot%7B%5Cphi%7D_2%28%5Cvec%7Bx%7D%27%2Ct%29%5D+&bg=ffffff&fg=808080&s=0 "[\phi_1(\vec{x},t),\dot{\phi}_2(\vec{x}',t)]")
…which is something that I wish would disappear, mawfuckas.
I mean, is this a condition of quantization? Perhaps.
![[\phi_r(\vec{x},t),\pi_s(\vec{x}{\,}',t)] = i \delta^3(\vec{x}-\vec{x}{\,}')\delta_{rs}](http://s0.wp.com/latex.php?latex=%5B%5Cphi_r%28%5Cvec%7Bx%7D%2Ct%29%2C%5Cpi_s%28%5Cvec%7Bx%7D%7B%5C%2C%7D%27%2Ct%29%5D+%3D+i+%5Cdelta%5E3%28%5Cvec%7Bx%7D-%5Cvec%7Bx%7D%7B%5C%2C%7D%27%29%5Cdelta_%7Brs%7D+&bg=ffffff&fg=808080&s=0 "[\phi_r(\vec{x},t),\pi_s(\vec{x}{\,}',t)] = i \delta^3(\vec{x}-\vec{x}{\,}')\delta_{rs}")
Realize that
and
…are independent [scalar] fields and will only be canonically conjugate in their own moments, denoted as
…on the left.
With that said…
![[\phi_1, \dot{\phi}_2] = 0](http://s0.wp.com/latex.php?latex=%5B%5Cphi_1%2C+%5Cdot%7B%5Cphi%7D_2%5D+%3D+0+&bg=ffffff&fg=808080&s=0 "[\phi_1, \dot{\phi}_2] = 0")
All you need for a vector space to exist is for the transformations to be defined such as a set of vector addition and scalar multiplication operations for RGB numbers. RGB numbers are physically bound within a range, however, if you were to measure x, y, z coordinates of Earth, some of the values that you end up with will be invalid. Also take into consideration that the mathematical concept of a vector space isn’t all that much different from that of a computer science one, at that. All the mathematics requires is that you have defined addition and multiplication that has eight axioms. Remember that axioms are axioms. Once you’ve defined addition and multiplication, they can be used to make statements that are either physically true or false. But, just because you can make a statement that “…the scalar multiplication of red multiplied by 2 will get us outside a set of physically valid colors” means that you have defined the axioms. The rules of vector spaces are that the mathematical operations are defined but necessarily that they always lead to physically plausible results. You can count the number of pine trees with integers. It’s plausible to talk about one trillion pine trees when it’s a fact that there aren’t that many trees in the world and that does not invalidate the use of number to describe pine trees. I haven’t seen a “negative” pine tree and neither have any of you but it is indeed plausible to define an additive inverse (for those of you that have read my blog post on “Watson’s ‘i/j’ Theory”, you now have a better understanding of why I use a negative number (-1) as an implied value for the diacritic dots over a lowercase i/lowercase j).
–Desmond
References
Title: “On The Differential Geometry of Curves In Minkowski Space”
Authors: J.B. Formiga, C. Romero
Date: Saturday, December 31st, 2005
Abstract: We discuss some aspects of the differential geometry of curves in Minkowski space. We establish the Serret-Frenet equations in Minkowski space and use them to give a very simple proof of the fundamental theorem of curves in Minkowski space. We also state and prove two other theorems which represent Minkowskian versions of a very known theorem of the differential geometry of curves in tridimensional Euclidean space. We discuss the general solution for torsionless paths in Minkowki space. We then apply the four-dimensional Serret-Frenet equations to describe the motion of a charged test particle in a constant and uniform electromagnetic field and show how the curvature and the torsions of the four-dimensional path of the particle contain information on the electromagnetic field acting on the particle.
