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Are Scalar Quantities in Physics really 1-D Affine Spaces in disguise?

  1. May 22, 2012 #1
    In Physics it seems quantities are either Scalars or Vectors (let's not get into tensors, but if there are other types, please tell me). Vectors I understand reasonably well. Scalars, however, I'm not so sure about:

    Consider the Scalar quantity of Temperature. There is absolute zero, but there is also a scale that sets zero at the freezing point of water. The conversion between the two requires a translation of the zero point. The unit degree also changes and is accounted for a scaling factor. There are no rotations in 1-D, but there is orientation and an interval on a temperature scale can be positive or negative. It appears to be a space with 1-D vectors defined as temperature changes from each point on the scale (formally: a vector space over the field of reals) and meet the definition of a 1-D affine space.

    Now consider mass. Mass doesn't have any standard scales whose zero point is offset from absolute zero mass. But wait, what if I rent a scale to weigh my gold and the proprietor sets the reading to zero for the first two grams as payment for my using his scale. It's not standard, but my take home gold after certifying its weight is 2 grams less (because I pay it to him). The conversion process is now the same as for the temperature scales: a translation is involved. Furthermore, he could use grams while I use ounces which I must account for with a scale factor. Does the potential for arbitrarily changing the zero on the mass scale mean that mass is also an affine space? Here we may also define a mass gain or loss as a 1-D vector quantity as we have done with temperature.

    I believe we can apply the same logic to potential energy (zero relative to local minimum), luminous intensity (zero is the point of detectability by the human eye), charge (set zero as the amount needed to make my hair stand on end). Sounds silly, but zero on the scales for these quantities is arbitrary. Does this make each more properly modeled as a variable that represents any member of an affine space, rather than just thinking of them naively as Reals (formally a Field with additional axioms in place to guarantee ordering among the members of the space)?

    Please be as mathematical as you want with your answers.
     
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  3. May 22, 2012 #2

    AlephZero

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    I think the important feature of a scalar quantity is that once everybody has agreed on a scale to measure it, its numerical value is the same in all coordinate systems.

    The numerical values of the components of a vector are different in different coordinate systems, but there is a simple "rule" for transforming the components between different systems So in that sense we can say there is "one thing" called a vector, even though different observers using diifferent coordinate systems would use different numbers to describe it.

    The same principle is true of higher order tensors. Stress and strain are familiar examples of second order tensors, and the properties of a general anisotropic material in continuum mechanics is a fourth order tensor (but with a lot of symmetry properties, so only 21 of its 81 elements are independent of each other).

    I would argue that tensors are the "basic" concept here - scalars and vectors are just zeroth and first order tensors.
     
  4. May 22, 2012 #3
    You have recognized the set of real numbers as a one dimensional vector space over the reals. In particular, you are singling out the addition structure and scaling structure of a real variable, which is the starting point of vector spaces.

    I don't see much else there is to your line of reasoning.

    (I hope you don't confuse my candor with impudence).
     
  5. May 22, 2012 #4
    Thank you both for your replies. I'm going to have to think about the tensor answer to see if it is what I am looking for. If it is, it will take a large change in my basic thinking.

    As for Reals being a vector space, I cannot quite agree. The Reals is a Field with extra stipulations such as upper bounding in order to guarantee that the members of the Field are ordered, but is not in and of itself a vector space. Reals are often used as Scalar coefficients in a vector space. A 1-D vector space can be defined over the Field of Reals (but it doesn't have to be Reals, it could be some other Field) and this vector space is appropriate for such things as showing the velocity in 1-D, as for example in a drag race. Nonphysically, the Field of Reals themselves can be assigned as the vectors in this 1-D vector space with the real number "1" designated as the basis vector to create a vector space of Reals over a Field of Reals.

    An Affine space has values that are intrinsic, and in 1-D, the race track with meter markers would be a good example. There is a mapping (interaction) from a 1-D Affine space that takes a point from the Affine set and adds a vector from the displacement vector space (for the racetrack example) at that point to yield another point within the Affine space. This Affine space retains the ability to have its origin changed, whereas the Reals do not.

    To extend this further, the meter is the basis vector in the displacement vector space and since in 1-D, the metric of a vector space is intrinsic, the meter also acts as the unit length and brings physical meaning to the vector space, which in turn is inherited by the Affine space. Once the origin in the Affine space is chosen, the displacement vector space at that point, (that is, the Real scalar coefficients from the displacement vector space at the origin) provides the meter markers for the entire space (racetrack in this case).

    I believe that time, position (the only one having 2 or more dimensions), mass, temperature, energy, charge, luminous intensity, etc are Affine in nature and that vectors are defined as intervals in these spaces. The unit length, unit mass, unit energy, unit temperature, etc. from these vector spaces provide the physical meaning of these quantities to their respective Affine spaces.

    I don't see this talked about, that is why I'm asking for other people's opinions.
     
    Last edited: May 22, 2012
  6. May 22, 2012 #5
    This isn't my opinion. I encourage you to work through the vector space axioms where the Abelian group is [itex]\mathbb{R}[/itex]and the field is [itex]\mathbb{R}[/itex].


    What do you mean by intrinsic? It seems to me that it's quite arbitrary in your examples.

    Here you are first treating elements in the vector space as a point, and then adding a displacement vector (which is again a member of the vector space).

    I again assert that your confusion stems from the fact that [itex]\mathbb{R}[/itex] is a one dimensional vector space.
     
  7. May 23, 2012 #6

    Stephen Tashi

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    The meaning of the statement "The Reals are a vector space" in mathematics is that the Reals statisfy the assumptions of a vector space. To say "The Reals are a vector space" does not mean that every vector space must satisfy the properties of the real numbers. You are apparently using the phrase "The Reals are a vector space" in some philosophical or poetic context, Mathematicians, Philosophers and everyone else use the word "are" and other forms of the verb "to be" ambiguously. The use of "are" in "The Reals are a vector space" is not the same as the use of "are" in "Those are the men who robbed me".
     
    Last edited: May 23, 2012
  8. May 23, 2012 #7
    Thank you for clarifying that point. I never got that subtlety before. That is a statement I quite agree with (I hope I used the word "is" properly here - LOL :-). I suppose I was being pedantic in my distinction.

    I assure you I have worked through the axioms and know them (for Fields and Vector Spaces) very well. I believe we are actually on the same page. It all comes down to a difference between the multiplication op in the Field of Reals and the Scaling op in the Set of Vectors. In the Field, the op is a closed mapping between two members within the same Set, whereas in the Set of Vectors, the op is a mapping between a member of the Set of Vectors and a member from the Set of Scalars (a different set entirely) that is closed within the Set of Vectors. The question arises: if we now let the Set of Vectors become one and the same with the Set of Scalars (i.e., the Field of Reals), do we indeed still have a Vector Space that is also a Field or do we just have a Field? I am uncertain of the answer as all Vector expressions would resolve to a single numerical value, collapsing the internal mechanical workings that make a Vector Space useful as a model for Physical processes.

    We all here understand Vector Spaces and the above question is interesting enough to find out other people's opinion. I believe it would be best to open a new thread on this topic as it is a peripheral issue to my question about Affine Space.

    I too am unhappy with the word intrinsic to describe Affine Quantities. There are many words I considered (Direct, Global, Absolute, Raw, Pure or Purebred, Unadulterated, Independent, Actual, Fundamental or Basic, Atomic or Indivisible, Non-derived, Simple). What I mean is that in an Affine Space, the quantities are directly measurable and they are absolute measurements because they are not relative to any other point in the space (although some may say it is "relative to a global origin." But that is actually a difference quantity of the same numerical value as the absolute quantity, so it is not really the same. Also it only works once the origin is chosen and the numerical values will become different if the origin is changed afterwards). The quantities are basic and pure in that they are not built upon other physical quantities and cannot be broken down (e.g., absolute location is in meters and is pure and underived, whereas a displacement is a derived quantity that is also measured in meters (actually delta_meters), but is relative to some other absolute location and is a difference between the two). The only reason there is a connection between Affine quantities and displacement vector quantities (in the example where a displacement is relative to the chosen origin) is because it is conventional to use the numerical labels from the Vector Space (displacement vector space for this example) associated with the origin as labels on the Affine points. Actually, they are the Real coefficients in the Vector Space that quantify the number of basis vector displacement units that are used to label the Affine points (so it really isn't a coincidence). So even though the Axioms and the construction of an Affine Space are far different than those of the Reals or of a Vector Space, the numerical labels on the Affine points are the same as for the Reals and the distinction between the two spaces should not be lost because of it.

    This choice of labels is not imperative. To be philosophical here, Affine points are pure because the label is secondary, it is the specific Affine point behind the label (and associated with the label) that is the important information to be ascertained in a physical measurement.

    No. Affine Sets do not have addition as a closed op and are not a Group on addition. They have a mapping between a member from the Affine Space (also called a point) and a member from a Vector Space (that is associated with the Affine member, but is still a completely different and separate Set) that is closed on the Affine Set. This is the relationship of a coordinate plane with location (Affine point) changed by adding a displacement (Vector quantity) to become some other location.

    No. That fact is not the source of my confusion. I'm not confused. It isn't central to the question. My question is whether or not a measurable physical quantity that is a rank 0 tensor (mass, temperature, energy, speed, charge, location, time, distance (magnitude of displacement vector), luminous flux, etc is properly modeled as an Affine Space. An Affine Space is not a 1-D Vector Space and if we compare it with the Field of Reals, a 1-D Vector Space over the Reals, or the Field of Reals as a Vector Space over the Reals as alternative models, we have very clear choices. Maybe it is none of the above. Then what would it be?
     
  9. May 23, 2012 #8
    Let me focus on this question then. It seems like you have narrowed your search of physical quantities to those whose measurements are real numbers. Fine, this puts the stipulation that a possible model via affine spaces requires the affine set to be the set of real numbers. For the translation properties, you need a vector space compatible with your affine set. The most sensible choice is the vectorspace ##\mathbb{R}## as it can take any point in the affine set to any other point.

    This is all fine, but it is equivalent to ##\mathbb{R}## in the sense of a vector space or field.

    So I believe the answer to your question is yes, you may choose to model your physical quantities as affine sets because you may view ##\mathbb{R}## as an affine set. However, the basic structure does not change, only your interpretation.

    I ask you a question then. Typically one makes a change of interpretation of something well known (e.g. viewing the real numbers as a subset of complex numbers) when there is something to be gained, such as a greater unification in research. Without viewing the reals as a subset of the complex numbers, one would not have the tools of contour integration to solve problems stated solely in terms of the reals. So what is the advantage of viewing the reals as an affine space? I can see some advantage to the interpretation that the reals are a vector space (or a tensor) because then it is part of the general theory of vector analysis. I say this because you mentioned you were happy with vectors. Do you want to discover some greater unity with more generalized objects? and if so how do these general objects relate to your interests?
     
  10. May 23, 2012 #9
    Your instincts are good. I am leading up to something bigger.

    Long goal: Fully Understand Universe
    Medium goal: Truly grasp the math and physics of String theory, Diff Geometry, SR, GR, Geometric Algebra, Quantum Theory, Thoery of Everything so that it is intuitively obvious and apparent to me on casual observation and even when I'm just thinking about it. I want it to be child's play: as easy as 1, 2, 3, like sesame street, or counting apples at the grocery store.
    Short goal: I want to fully understand the nitty gritty of all the fundamentals. When I began looking at tensors, I could see they were not intuitive at all, so I assumed I'm missing something. I began looking at Vector Spaces and realized something was missing. So I began looking at Affine Spaces and I am now trying to fit all the pieces together.

    I believe there is a unified mathematical structure that binds (in Physics jargon) Vectors and Scalars. Vectors are abstract basis units that we quantify explicitly with Real coefficients (let's forget the complex case for now). An example is the velocity of a projectile in 3-Space. The abstract basis units in x, y, and z could be, say meters/sec. Two important issues here: 1 - it's really delta_meters/sec and it is confusing to say it the other way, and 2 - the basis unit is an intuitively understood concept we have wired into our brains by learning about it.

    The first issue tells me that there is a space somewhere that has more basic units of meters and not delta_meters and I want to know how they connect. The second issue tells me that the measurement unit is hidden and inaccessible and we must be comfortable with the coefficients as the representative of that unit when we deal with it mathematically. These are ideas I have come across on my journey. It turns out that the trio of spaces: Affine Space associated with a Vector Space over a Field of Reals tells the fuller story.

    There are many physical properties associated with a projectile. Its location (in meters) I find in an Affine Space. Location is the only property where the space can be multi-dimensional. In my example above it is 3D. With the origin chosen, the space of Displacement Vectors there connects the origin with transport to any other location in the Affine Space. Furthermore, it is the Metric defined in this Vector Space that integrates the 3 1-D Affine location spaces together into a seamless 3D whole (it also integrates the 3 1-D Vector Spaces into a 3D whole) and the choice of Basis Vectors (delta_meters in each of the 3 directions) that provide the Affine Space with meaningful physical definition of length and units of measure.

    The Reals are where the rubber meets the road. The Basis Vectors are abstract, but their coefficients tell us quantity, once we agree on units of measure (which are the Basis Vectors). The Affine locations are labeled with appropriate coefficients that the Affine Space takes from the Displacement Vector Space. Now, change the Basis and Metric with a linear transformation of the Vector Space and the Affine Space changes automatically. I like this unification. Change the origin, and the Displacement Vector Space is changed correspondingly. Now the labels in the Affine space are updated appropriately according to the Displacement Vector Space at the new origin. All the gears and mechanics make sense mathematically and physically. Now, time is 1-D, but the same underlying machinery applies. I want to be able to say that all basic properties of the projectile can be modeled exactly the same. Temperature is often described as an Affine quantity. I would like to think mass, charge, and all the others that Physicists call Scalars are also. Changes in any Affine quantity would be Vector quantities. Location (in meters) + Displacement (in delta_meters) = New Location (in meters). Mass (in kg) + Gain in Mass (in delta_kg) = New Mass (in kg).

    This also reduces confusion about units in calculations because to me Vectors or changes in Affine quantity must use delta_units to be perfectly correct. They should become part of the formal doctrine. Something Physicists have overlooked. I want to be aware of these pitfalls and sources of confusion.

    This also clears up confusion about the use of arrows and direction/magnitude descriptions of Vectors. Arrows are Vectors depicted in Affine Space. If they are Displacement or Change Vector Spaces, the length of the arrows is exact in the depiction. If they are other relevant Vector Spaces such as Velocity Vector, Acceleration Vector, or any other Vector Space that have location dependence (they correspond to the individual terms in a taylor series expansion of the projectile path), the arrows are proportional to each other, but have no explicit connection to the scale of the Affine Space's depiction.

    The Vector Spaces themselves, when depicted systematically on a Cartesian grid with the Real coefficients to the Basis Vectors labeling each grid point, do not contain arrows. We can pretend that they do, but each point in the description is representative of a Vector, not a location point. We can pretend that the arrow from the zero Vector to some point in the Vector Space depiction represents the Vector, but it doesn't. It also explains why Displacement Vectors in depictions of the Affine Space are freely mobile, but they are not in the depictions of the Vector Space. You would be amazed at the percentage of physicists that have no concept that they have confused Vector vs Affine Space and who do not know when the use of an arrow (or direction/magnitude description of a Vector) is and is not correct.

    Moving on. Once I clear up this question, I want to begin casting the laws of physics in well defined mappings between members of different Affine Sets and different Vector Spaces which map into members of other Affine or Vector Sets. For example F = ma is a mapping from the Mass Affine Space and Acceleration Vector Space into a Force Vector Space. Then I want to revisit Tensors and make sense of them.

    Does this answer your question?
     
    Last edited: May 23, 2012
  11. May 23, 2012 #10
    There are two remarks I will make. Do with them as you like.

    You are using abstract mathematical concepts and trying to prescribe a physical description on them. A vector must this,a point must be this, etc. This is counter to purpose of generalization, which is to unify. When a mathematician thinks of a vector, he thinks of it as it's own entity, free of the laws of physics. It just so happens that they turn out to be useful in certain cases. That is, they are adaptable. For example, QM finds it's mathematical home in terms of functional analysis, which in practice deals with vector spaces whose vectors are functions. You have chosen a few properties of physical space in which you take to be the most important, and have decided that affine space describes them best. This may be true, but that does not mean that affine spaces should naturally be associated with your descriptions.

    Secondly, and this is even more subjective than the previous, I think it's a very bad idea to learn math 'from the ground up'. If you feel that tensors are not intuitive, climb up, not down. Learn the best you can in how tensors are used, theorems about tensors, generalizations of tensors, etc. I have found that deeply learning advanced mathematics sheds much greater insight on more basic math, than the reverse direction. It's not important to learn the exact construction of the real number system (via completion of the rationals) as it is to get your hands dirty with real numbers first. In fact, learning basic completion properties of metric spaces sheds much light on the reals.

    OK I fibbed, there is a third remark I have. Your goal is to understand certain topics intuitively. Plainly, it is not the case that once you build intuition for something, you no longer need to refer back. In all likelihood, you will come back again and again to refine your intuition. Even then, your intuition likely will not match someone else's, and they will be just as correct as you are yourself.

    You should probably preface everything I said with my opinion, I find actually writing it is overly distractive and besides the point.
     
  12. May 24, 2012 #11
    Rising Eagle,
    I think post#10 offers some useful advice to which I would like to add the following cautions.

    It is unfortunate that different scientific disciplines sometimes have different definitions for terms. Vector is one such term where there are at least five different definitions in use.

    1)These are the strict higher mathematical definition as an element of a vector space. This is also used in advanced physics. This definition does not require a binary multiplication operation.

    2)The definition used in higher physics and engineering vector calculus with a non commutative binary operation leading to grad, div, curl, pseudo vectors or axial vectors. Chirality, of which handedness is a simple example is improperly treated in this system.

    3)The definition used in advanced physics with a more advanced form of binary product (hodge) that fully treats chirality in the real world.

    4)The definition used in biological and medical science as a carrier.

    5)The definition used in computer science as an row or column element of a table.

    go well
     
  13. May 24, 2012 #12
    Hi all,

    I would like to offer a couple points on the difference between quantities that are "fundamentally" affine vs. not affine in nature. In mathematics, an affine space is a space whose group of symmetries includes the translations of a vector space. A physicist would know better, but I can't think of a single quantity in physics that is truly, surely known to have that kind of symmetry (not even time, according to the Big Bang). So lets idealize the universe so that we have examples.

    The classical universe is a 3D x 1D affine space in which R3 X R1 acts by translation. According to this model, every point in space is fundamentally the same as every other because there is a symmetry which carries one point to the other. There are more symmetries such as rotation and inertial changes in coordinates (modulo relativity), but those do not directly bear on affine vs not affine.

    Many of the quantities you mentioned do not have this property at all. Kinetic energy is not affine because the range of acceptable values form a ray, not a line. KE=0 is physically different from KE=1. There is a kind of scaling symmetry to the universe that renders KE=1 physically the same as KE=2, but in order to realize that symmetry you have to change your scale of mass or length or time. But there is never any symmetry of the universe that will make KE=0 equivalent to KE=1. And we could say the same thing for mass or temperature. Going back to the geometric definition of affine, you cannot arbitrarily translate a point on a ray and still end up with a point on the ray.

    Finally, it is obviously true that R satisfies the definition to be a vector space. But then so does a tensor space. But it would be highly misleading to refer to an element of a tensor space as a vector. I say call it as it is. Mass/Kinetic energy/temperature/speed are numbers, not vectors (unless they are in fact vectors in modern physics :smile:).
     
  14. May 24, 2012 #13
    Thank you rising eagle for an interesting thread.

    I am learning from what others have written besides (I hope) contributing something myself.

    Vargo

    Why do you say this?
     
  15. May 24, 2012 #14
    The mathematician's reason is that it is just plain confusing. If I said this tensor is a vector, then everyone familiar with tensors would assume that the tensor is of type (1,0).

    The physical reason is that quantities in nature are characteristically of a certain type, and "vectors" are fundamentally related to spatial displacements. If I am talking about displacements or velocities along a line, then I am talking about vectors. If I am talking about speed or kinetic energy, then I am talking about a number.

    Perhaps we could agree to the following convention. From now on, what are now called Vector spaces, we shall henceforth call Linear spaces. And what are now called vectors, we shall call.... um....... Linear combinationables (combinables?) :). The term vector shall exclusively mean exactly what we thought it meant before we learned about linear algebra.
     
  16. May 24, 2012 #15
    So this is really by way of a suggestion - not a statement of current practice?

    By the way affine and linear are different.
     
  17. May 24, 2012 #16
    I would say it is more or less a statement of current practice and not a statement about current strict logical definitions. For example, if a speaker is talking about a Hilbert space related to a PDE, then she might refer to it as a vector space, but she would not be likely to refer to its elements as vectors. Rather they would be referred to as functions.

    Actually in variational problems, the Hilbert space often comes up as a space of displacements from some base function, so they are like infinite dimensional vectors. But still, we call them functions anyway because it is specific to the problem at hand.

    Re: linear vs. affine. Yes I know. I didn't mean to suggest otherwise.
     
  18. May 24, 2012 #17
    If you were to say that the terminology has become confused I would agree with you.
    That is what I pointed out in post #11. In particular uses 4 and 5 do not even conform to the vector space axioms.

    In particular I think the term vector space should be renamed. But that's not going to happen.

    I think you would have problems restricting vectors to the geomteric vectors you like simply because these require an additional binary operation, not part of the axiom set, to complete their definition. In addition (pun not intended) you have to face the problem of improper rotations.

    Whilst most would certainly refer to the solutions of pdes as functions, where would you place functional analysis in your scheme of things?
     
  19. May 24, 2012 #18
    What do you mean when you say that geometric vectors require an additional binary operation? Are you referring to a metric? If I had to define what I mean by vector, I would say the elements of the tangent space of a manifold (or some technical extension of that to algebraic varieties/non-smooth manifolds).

    Re: functional analysis. Scheme is a strong word for my internet ramblings in between semesters :smile:. My only scheme is to try to refer to things in the most specific way possible for the problem at hand.
     
  20. May 24, 2012 #19
    The vector space axioms include the binary operation of vector addition of two elements of the space as necessary.

    The binary operation of multiplication of two elements (vectors) is not part of the axiom set {either interior (dot) or exterior (cross) products}.
    Some vector spaces have one or both, some have neither.
    You require both for the vector calculus used by physicists in abundance.
     
  21. May 24, 2012 #20
    Ah I see what you mean. I guess this just serves to illustrate the futility of isolating exactly what is a vector. Perhaps a mathematician's "vector" is not a physicists.

    Here is a physical example of a vector in which neither type of product plays a role. Consider a physical system consisting of two particles in space subject to a force that depends only on the distance between the particles. According to Newton's second law we get a system of 6 2nd order ODE. One for each component of the position of each particle. By introducing velocity or momentum, we can transform it into a system of 12 first order ODE. So we have a 12 dimensional phase space which is the set of all possible positions and velocities. And on that space, we have the 1st order differential equation.
    X'=G(X),
    The thing on the right side is a vector field, precisely because it tells us the infinitesimal displacements of X(t).
     
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